#475942

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

756 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=g,c^2=f,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=di,d^c=de,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, 383138);