#7062491

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1969 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)(5,6)(7,9)(8,10)(11,12)(13,14,16,22,17,21,25,19)(15,18,24,26,23,20,28,27);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,17,23)(14,18,21,20)(16,24,25,28)(19,27,22,26);(5,7,8,12)(6,9,10,11)(14,19)(15,23)(16,25)(18,26)(20,27)(21,22);(2,4)(7,12)(9,11)(13,15)(14,20)(16,24)(17,23)(18,21)(19,26)(22,27)(25,28);(6,10)(9,11)(14,21)(15,23)(19,22)(24,28)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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, 6969687);