#451099

order	512
nilpotency class	3
isoclinism	isoclinic class of #417454
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1245 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358295);