#7006525

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6913721);