#6806994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2039 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714190);