#294386

order	512
nilpotency class	3
isoclinism	isoclinic class of #267755
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×#392	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201582);