#479762

order	512
nilpotency class	3
isoclinism	isoclinic class of #478580
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2789
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17236
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

557 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,13,16,8)(4,9,20,32)(5,22,23,52)(6,12)(7,11)(10,24,35,53)(14,30)(15,42)(17,39,45,29)(18,27)(19,48)(21,33,25,37)(26,36)(28,43)(31,54)(34,50)(38,60)(40,61,47,64)(41,58)(44,51)(46,56)(49,59)(55,63,57,62);(1,3,6,17,7,16,26,45)(2,8,12,29,11,13,36,39)(4,18,21,46,20,14,25,41)(5,15,24,43,23,44,53,60)(9,30,33,58,32,27,37,56)(10,28,22,51,35,38,52,42)(19,40,50,62,49,47,54,63)(31,55,48,64,34,57,59,61);(1,4)(2,9,11,32)(3,14)(5,19)(6,25)(7,20)(8,27,13,30)(10,34,35,31)(12,37,36,33)(15,40)(16,18)(17,46)(21,26)(22,48,52,59)(23,49)(24,54)(28,57,38,55)(29,58,39,56)(41,45)(42,61,51,64)(43,63)(44,47)(50,53)(60,62);(1,5)(2,10,11,35)(3,15)(4,19)(6,24)(7,23)(8,28,13,38)(9,31,32,34)(12,22,36,52)(14,40)(16,44)(17,43)(18,47)(20,49)(21,50)(25,54)(26,53)(27,55,30,57)(29,51,39,42)(33,48,37,59)(41,62)(45,60)(46,63)(56,64,58,61)

Polycyclic presentation:

2^9|a^2=efg,b^2=h,c^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386958);