#7148207

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

1813 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)(15,20,30,32,27,35,26,34)(16,22,31,36,28,37,29,38)(25,33,41,42,39,43,40,44);(1,2)(3,4)(6,9)(7,11)(13,15,18,27)(14,20,24,35)(16,25,28,39)(17,26,19,30)(21,32,23,34)(22,33,37,43)(29,40,31,41)(36,42,38,44);(5,6)(7,10)(8,9)(11,12)(14,21)(17,19)(20,32)(22,36)(23,24)(26,30)(29,31)(33,42)(34,35)(37,38)(40,41)(43,44);(2,4)(6,9)(10,12)(13,16)(14,22)(15,25)(17,29)(18,28)(19,31)(20,33)(21,36)(23,38)(24,37)(26,40)(27,39)(30,41)(32,42)(34,44)(35,43);(5,7,8,11)(6,10,9,12)(16,28)(22,37)(25,39)(29,31)(33,43)(36,38)(40,41)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,e^2=h,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, 7055403);