#382267

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

968 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)(3,6)(4,5)(7,8)(10,13)(11,15)(17,18,22,20,21,25,23,32)(19,26,30,27,28,24,29,31);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,18,23,32,21,25,22,20)(19,27,30,24,28,31,29,26);(3,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,31)(21,28)(22,30)(23,29)(25,26)(27,32);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(17,20,22,25,21,32,23,18)(19,27,30,24,28,31,29,26)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=efh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289463);