#468051

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(17,18,21,24)(19,26,20,30)(22,32,31,25)(23,29,27,28);(2,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,23)(21,24)(22,25)(26,29)(28,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,19,21,20)(18,23,24,27)(22,29,31,28)(25,30,32,26);(3,7)(6,8)(17,20)(18,23)(19,21)(22,29)(24,27)(25,26)(28,31)(30,32)

Polycyclic presentation:

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

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, 375247);