#7145928

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2225 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)(7,11)(10,12)(13,14,18,22)(15,19,24,32)(16,21,27,36)(17,20,29,23)(25,31,37,43)(26,33,38,30)(28,34,40,35)(39,42,44,41);(2,3)(6,9)(7,11)(13,15,18,24)(14,19,22,32)(16,25,27,37)(17,26,29,38)(20,33,23,30)(21,31,36,43)(28,39,40,44)(34,42,35,41);(5,6)(7,10)(8,9)(11,12)(14,20)(15,24)(16,27)(17,29)(19,30)(21,35)(22,23)(31,42)(32,33)(34,36)(39,44)(41,43);(1,3,4,2)(6,9)(10,12)(13,16)(14,21)(15,25)(17,28)(18,27)(19,31)(20,34)(22,36)(23,35)(24,37)(26,39)(29,40)(30,41)(32,43)(33,42)(38,44);(5,7,8,11)(6,10,9,12)(14,22)(15,24)(16,27)(20,23)(26,38)(28,40)(31,43)(41,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053124);