#495933

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

608 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403129);