#6815181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1912 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722377);