#458716

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1063 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:

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

Polycyclic presentation:

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

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, 365912);