#475308

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=h,c^2=i,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,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, 382504);