#6514137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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
prime radical	#35237	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=fi,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421333);