#7149100

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

1935 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)(13,14,19,24,18,20,17,23)(15,25,27,33,28,38,32,21)(16,22,30,36,29,34,31,37)(26,39,42,43,40,44,41,35);(1,2,3,4)(6,9)(7,11)(14,20)(16,29)(21,33)(23,24)(25,38)(26,40)(30,31)(41,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,28)(14,21,20,33)(16,26,29,40)(17,27,19,32)(22,35,34,43)(23,25,24,38)(30,41,31,42)(36,44,37,39);(2,4)(6,9)(10,12)(13,16)(14,22)(15,26)(17,31)(18,29)(19,30)(20,34)(21,35)(23,37)(24,36)(25,39)(27,42)(28,40)(32,41)(33,43)(38,44);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,20,24)(15,27,28,32)(16,30,29,31)(21,25,33,38)(22,36,34,37)(26,41,40,42)(35,44,43,39)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,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, 7056296);