#328809

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ehi,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236005);