#6816228

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

1831 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)(15,24)(17,25)(19,28)(21,26);(5,7,8,12)(6,9,10,11)(13,14,18,22)(15,25,24,17)(16,20,27,23)(19,21,28,26);(1,2,3,4)(6,10)(7,12)(13,15,16,26,18,24,27,21)(14,19,23,25,22,28,20,17);(6,10)(9,11)(13,16,18,27)(14,20,22,23)(15,21,24,26)(17,19,25,28);(13,17,27,28,18,25,16,19)(14,21,20,24,22,26,23,15)

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,d^2=i,e^2=fhi,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, 6723424);