#349808

order	512
nilpotency class	3
isoclinism	isoclinic class of #346294
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×#392	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1176 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)(3,6)(4,5)(7,8)(10,14)(11,12)(15,16)(25,31)(28,32);(2,5)(6,8)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,28,31,32);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,31)(30,32);(9,11,13,12)(10,14,15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257004);