#454295

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,d^c=de

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361491);