#475261

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

820 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)(9,10,13,12)(11,14,15,16)(17,18,21,30,22,24,31,26)(19,27,20,23,29,25,28,32);(2,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,20,22,29,31,28)(18,23,26,27,24,32,30,25);(1,3)(2,6)(4,7)(5,8)(11,15)(14,16)(18,24)(19,28)(20,29)(21,31)(23,27)(25,32);(3,7)(6,8)(9,12,13,10)(11,16,15,14)(17,20,21,29,22,28,31,19)(18,25,26,23,24,27,30,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi

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, 382457);