#304252

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

1090 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=ef,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, 211448);