#7064336

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

1936 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971532);