#6810842

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1922 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718038);