#6682914

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

2031 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)(15,20)(16,21)(17,26)(18,27)(19,23)(22,24)(25,28);(5,7,8,12)(6,9,10,11)(13,15,19,26)(14,17,23,20)(16,21,25,28)(18,27,24,22);(1,2,3,4)(13,16,19,25)(14,21,23,28)(15,24,26,18)(17,27,20,22);(6,10)(9,11)(13,17,19,20)(14,15,23,26)(16,27,25,22)(18,21,24,28);(7,12)(9,11)(13,18,19,24)(14,22,23,27)(15,25,26,16)(17,28,20,21)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590110);