#499634

order	512
nilpotency class	4
isoclinism	isoclinic class of #499261
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3184
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23781
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

435 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15,18,35)(4,10)(5,12)(6,27)(7,14)(8,11)(9,17,33,20)(13,30)(16,47,50,45)(19,41,52,46)(21,54,56,63)(22,44)(23,37)(24,58)(25,39)(26,43)(28,61,51,34)(29,60)(31,40)(32,48,62,49)(36,55,64,53)(38,59)(42,57);(1,3)(2,9)(4,21)(5,20)(6,28)(7,17)(8,18)(10,36)(11,33)(12,35)(13,41)(14,15)(16,24)(19,31)(22,55)(23,53)(25,56)(26,52)(27,46)(29,49)(30,51)(32,38)(34,43)(37,63)(39,64)(40,61)(42,45)(44,54)(47,60)(48,57)(50,59)(58,62);(1,4)(2,10)(3,16,18,50)(5,23)(6,29)(7,22)(8,25)(9,32,33,62)(11,39)(12,37)(13,42)(14,44)(15,45,35,47)(17,49,20,48)(19,53,52,55)(21,28,56,51)(24,26)(27,60)(30,57)(31,59)(34,63,61,54)(36,41,64,46)(38,40)(43,58);(2,11)(4,22,25,23)(6,30)(9,33)(10,37,39,44)(12,14)(15,35)(16,48,50,49)(19,52)(21,55,56,53)(24,57,59,29)(26,31)(28,51)(32,45,62,47)(36,63,64,54)(38,42,58,60)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=eh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283942, C₂×#499634
maximal normal subgroup of	#72283942, C₂×#499634
maximal quotient of	C₂×#499634

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406830);