#7006285

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

2065 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)(13,14,18,22,17,19,30,34)(15,23,26,32,24,38,31,20)(16,21,29,37,27,33,41,44)(25,39,40,36,28,42,43,35);(2,3)(5,6)(7,11)(8,10)(9,12)(14,19)(15,24)(16,27)(22,34)(26,31)(29,41)(35,36)(39,42);(5,7)(6,9)(8,12)(10,11)(13,15,17,24)(14,20,19,32)(16,28,27,25)(18,31,30,26)(21,36,33,35)(22,38,34,23)(29,40,41,43)(37,39,44,42);(1,3,4,2)(7,12)(9,11)(13,16)(14,21)(15,25)(17,27)(18,29)(19,33)(20,35)(22,37)(23,39)(24,28)(26,40)(30,41)(31,43)(32,36)(34,44)(38,42);(6,10)(9,11)(14,19)(16,27)(20,32)(22,34)(23,38)(25,28)(29,41)(40,43)

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,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, 6913481);