#7614492

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521688);