#454543

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, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1712 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)(5,6,10,9)(7,12,8,11)(13,14,17,20)(15,21,23,18)(16,19,24,27)(22,25,28,26);(5,7,10,8)(6,11,9,12)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,26,27,25);(5,8,10,7)(6,11,9,12)(13,16,17,24)(14,19,20,27)(15,22,23,28)(18,25,21,26);(2,4)(5,9,10,6)(7,12,8,11)(16,24)(19,27)(22,28)(25,26)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=ei,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, 361739);