#7149204

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

2041 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056400);