#6520358

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500434
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2619 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427554);