#6814484

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

1988 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,16,21,20)(17,26,27,23)(19,24,25,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,22)(17,25)(18,21)(19,27)(23,28)(24,26);(1,2,3,4)(13,16,25,23,18,20,19,26)(14,21,28,17,22,15,24,27);(6,10)(9,11)(14,22)(16,20)(23,26)(24,28);(7,12)(9,11)(13,17,19,21,18,27,25,15)(14,23,24,16,22,26,28,20)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,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, 6721680);