#6682946

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

2166 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)(15,22)(20,27)(23,28)(24,26);(5,7,8,12)(6,9,10,11)(13,14)(15,23)(16,18)(17,19)(20,24)(21,25)(22,28)(26,27);(1,2,3,4)(14,18)(16,25)(19,21)(20,23)(24,26)(27,28);(6,10)(9,11)(14,19)(15,22)(18,21)(24,26);(7,12)(9,11)(13,15,17,22)(14,20,19,27)(16,26,25,24)(18,23,21,28)

Polycyclic presentation:

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

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, 6590142);