#7611543

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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

2138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,3)(5,6)(7,9)(8,10)(11,12)(15,23)(20,24)(25,28)(26,27);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,14,18,22)(15,24,23,20)(16,17,21,19)(25,26,28,27);(6,10)(7,11)(13,15)(14,20)(16,25)(17,27)(18,23)(19,26)(21,28)(22,24);(6,10)(9,12)(13,16,19,14,18,21,17,22)(15,25,26,20,23,28,27,24);(1,3,4,2)(13,17,18,19)(14,21,22,16)(15,26,23,27)(20,25,24,28)

Polycyclic presentation:

2^9|b^2=fi,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7518739);