#434058

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
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	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1188 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)(5,6,9,8)(7,11,12,10)(13,14,17,19)(15,23,25,38)(16,27,28,37)(18,32,31,22)(20,35,33,26)(21,29,34,30)(24,36,40,44)(39,41,43,42);(1,3,4,2)(5,7,9,12)(6,10,8,11)(14,19)(16,28)(20,33)(22,32)(23,38)(24,40)(29,30)(41,42);(2,3)(5,8,9,6)(7,11,12,10)(13,15,18,26,17,25,31,35)(14,20,22,23,19,33,32,38)(16,24,29,41,28,40,30,42)(21,36,37,39,34,44,27,43);(2,3)(7,12)(10,11)(13,16)(14,21)(15,24)(17,28)(18,29)(19,34)(20,36)(22,37)(23,39)(25,40)(26,41)(27,32)(30,31)(33,44)(35,42)(38,43)

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=efi,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, 341254);