#498763

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2380
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23355
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 194 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cef,d^b=de,f^b=fi,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405959);