#416359

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6,9,10)(7,11,8,12)(13,14)(15,24)(16,28)(17,32)(18,23)(19,22)(20,26)(21,31)(25,41)(27,37)(29,44)(30,39)(33,36)(34,40)(35,42)(38,43);(6,10)(11,12)(13,15,18,33,19,27,17,26)(14,20,32,37,22,36,23,24)(16,29,34,42,30,43,31,25)(21,38,39,35,40,44,28,41);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,31,19,30,18,34)(14,21,23,28,22,40,32,39)(15,25,26,43,27,42,33,29)(20,35,24,38,36,41,37,44);(5,8,9,7)(6,11,10,12)(14,22)(16,30)(20,36)(23,32)(24,37)(25,42)(29,43)(31,34)

Polycyclic presentation:

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

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, 323555);