#6685342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1931 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:

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

Polycyclic presentation:

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

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, 6592538);