#7142947

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

2015 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)(13,14,18,23)(15,19,26,34)(16,27,28,21)(17,20,30,22)(24,37,38,32)(25,31,40,33)(29,36,41,35)(39,43,44,42);(1,2)(3,4)(6,9)(7,11)(13,15,18,26)(14,19,23,34)(16,24,28,38)(17,25,30,40)(20,31,22,33)(21,32,27,37)(29,39,41,44)(35,42,36,43);(5,6)(7,10)(8,9)(11,12)(14,20)(17,30)(19,31)(21,35)(22,23)(25,40)(27,36)(29,41)(32,42)(33,34)(37,43)(39,44);(2,4)(6,9)(10,12)(13,16)(14,21)(15,24)(17,29)(18,28)(19,32)(20,35)(22,36)(23,27)(25,39)(26,38)(30,41)(31,42)(33,43)(34,37)(40,44);(5,7,8,11)(6,10,9,12)(16,28)(21,27)(24,38)(29,41)(32,37)(35,36)(39,44)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050143);