#7149170

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
prime radical	#42648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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)(13,14,19,21,18,24,17,23)(15,20,30,35,25,34,27,32)(16,22,31,36,28,38,29,37)(26,33,41,43,39,44,40,42);(1,2,3,4)(6,9)(7,11)(13,15)(14,20)(16,26)(17,27)(18,25)(19,30)(21,35)(22,33)(23,32)(24,34)(28,39)(29,40)(31,41)(36,43)(37,42)(38,44);(5,6)(7,10)(8,9)(11,12)(14,21)(15,25)(16,28)(17,19)(20,32)(22,37)(23,24)(33,43)(34,35)(36,38)(40,41)(42,44);(2,4)(6,9)(10,12)(13,16,18,28)(14,22,24,38)(15,26,25,39)(17,29,19,31)(20,33,34,44)(21,36,23,37)(27,40,30,41)(32,42,35,43);(5,7,8,11)(6,10,9,12)(15,25)(16,28)(20,34)(22,38)(27,30)(29,31)(32,35)(36,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,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, 7056366);