#469422

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1114 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^c=hi,f^d=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, 376618);