#494790

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

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

636 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)(15,20)(16,22)(17,30)(18,34)(19,24)(21,26)(23,32)(25,35)(27,38)(28,37)(29,41)(31,36)(33,44)(39,43)(40,42);(1,3,4,2)(6,9)(10,12)(13,15)(14,20)(16,28)(17,26)(18,25)(19,27)(21,30)(22,37)(23,36)(24,38)(29,43)(31,32)(33,42)(34,35)(39,41)(40,44);(5,7,8,11)(6,10,9,12)(13,15)(14,21)(16,25)(17,31)(18,28)(19,27)(20,30)(22,36)(23,37)(24,40)(26,32)(29,42)(33,43)(34,39)(35,41)(38,44);(13,16,19,18)(14,22,24,34)(15,25,27,28)(17,29,33,32)(20,35,38,37)(21,36,40,39)(23,30,41,44)(26,31,42,43)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=i,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, 401986);