#460337

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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1315 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)(6,8)(9,10,12,14)(11,13,15,16)(19,28)(20,30)(21,22)(24,32)(25,26)(29,31);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18,23,27)(19,24,29,20)(21,25,22,26)(28,32,31,30);(3,7)(6,8)(11,15)(13,16)(17,19)(18,24)(20,27)(21,31)(22,28)(23,29)(25,30)(26,32);(1,3,4,7)(2,6,5,8)(17,20,23,24)(18,19,27,29)(21,30,22,32)(25,28,26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=ei,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, 367533);