#6811680

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24244	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2063 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,12)(9,11)(13,14,17,22)(15,24,26,19)(16,28,30,21)(18,20,32,23)(25,34,29,37)(27,35,40,33)(31,38,42,36)(39,43,41,44);(2,3)(5,7,10,6)(8,12,9,11)(13,15,17,26)(14,19,22,24)(16,29,30,25)(18,27,32,40)(20,33,23,35)(21,37,28,34)(31,41,42,39)(36,44,38,43);(1,2,4,3)(14,20)(18,32)(19,33)(21,36)(22,23)(24,35)(27,40)(28,38)(31,42)(34,43)(37,44)(39,41);(5,8,10,9)(6,11,7,12)(13,16,17,30)(14,21,22,28)(15,25,26,29)(18,31,32,42)(19,34,24,37)(20,36,23,38)(27,39,40,41)(33,43,35,44);(1,3,4,2)(5,9,10,8)(6,11,7,12)(14,22)(15,26)(16,30)(20,23)(27,40)(31,42)(34,37)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718876);