#7006422

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

2162 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)(13,14,15,22,17,21,25,18)(16,20,24,28,26,27,23,19);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,17,25)(14,18,21,22)(16,24,26,23)(19,27,28,20);(5,7)(6,9)(8,12)(10,11)(13,16,17,26)(14,19,21,28)(15,23,25,24)(18,20,22,27);(1,3,4,2)(7,12)(9,11)(13,16)(14,20)(15,24)(17,26)(18,19)(21,27)(22,28)(23,25);(6,10)(9,11)(13,15,17,25)(14,18,21,22)(16,23,26,24)(19,20,28,27)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6913618);