#6685609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

1965 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)(7,10)(8,9)(11,12)(14,18)(19,27)(20,25)(21,28);(6,9)(7,11)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,26,27,23);(1,2,3,4)(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21);(6,9)(10,12)(15,22)(19,27)(21,28)(23,26);(5,7,8,11)(6,10,9,12)(14,20)(17,24)(18,25)(19,28)(21,27)(23,26)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6592805);