#6814482

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

1966 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)(7,11)(8,10)(9,12)(13,14,19,18)(15,17,22,26)(16,20,24,25)(21,23,27,28);(5,7)(6,9)(8,12)(10,11)(13,15,19,22)(14,17,18,26)(16,23,24,28)(20,27,25,21);(1,2,3,4)(13,16,18,25,19,24,14,20)(15,21,17,23,22,27,26,28);(6,10)(9,11)(16,24)(20,25)(21,27)(23,28);(7,12)(9,11)(13,17,18,22,19,26,14,15)(16,23,25,27,24,28,20,21)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,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, 6721678);