#7149112

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

1942 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)(13,14,19,21,17,23,18,24)(15,25,32,34,26,20,28,36)(16,22,33,37,30,40,31,39)(27,38,42,43,29,35,41,44);(1,2,3,4)(6,9)(7,11)(13,15)(14,20)(16,29)(17,26)(18,28)(19,32)(21,36)(22,38)(23,25)(24,34)(27,30)(31,42)(33,41)(35,40)(37,43)(39,44);(5,6)(7,10)(8,9)(11,12)(14,21)(15,26)(16,30)(18,19)(20,34)(22,39)(23,24)(25,36)(35,44)(37,40)(38,43)(41,42);(2,4)(6,9)(10,12)(13,16)(14,22)(15,27)(17,30)(18,31)(19,33)(20,35)(21,37)(23,40)(24,39)(25,38)(26,29)(28,41)(32,42)(34,43)(36,44);(5,7,8,11)(6,10,9,12)(14,23)(15,26)(16,30)(21,24)(28,32)(31,33)(35,38)(43,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7056308);