#6685610

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

1919 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)(15,16)(19,28)(20,24)(25,26);(6,9)(7,11)(13,14,17,22)(15,24,16,20)(18,21,27,23)(19,26,28,25);(1,2,3,4)(13,15)(14,19)(16,17)(18,26)(20,23)(21,24)(22,28)(25,27);(6,9)(10,12)(13,16,17,15)(14,20,22,24)(18,26,27,25)(19,21,28,23);(5,7,8,11)(6,10,9,12)(14,21)(18,27)(19,24)(20,28)(22,23)(25,26)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,e^2=f,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, 6592806);