#6807965

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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

1864 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715161);