#349328

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1028 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,d^2=i,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh,g^c=gi,f^d=fi

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, 256524);