#7156167

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2067 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063363);