#294524

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1008 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)(3,7)(4,5)(6,8)(9,10,14,16)(11,12,15,13)(17,18,21,24)(19,27,28,23)(20,30,29,32)(22,26,31,25);(2,5)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,22)(21,28)(24,27)(25,30)(26,32)(29,31);(9,11,12,16,14,15,13,10)(17,20,22,28,21,29,31,19)(18,23,25,30,24,27,26,32);(1,3,4,8)(2,6,5,7)(18,24)(23,27)(25,26)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201720);