#6816155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

2330 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723351);