#438493

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

951 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:

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

Polycyclic presentation:

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

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, 345689);