#494033

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1026 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)(7,10)(8,9)(11,12)(13,14)(15,25)(16,22)(17,23)(18,21)(19,33)(20,27)(24,32)(26,40)(28,39)(29,35)(30,37)(31,38)(34,43)(36,42)(41,44);(6,9)(10,12)(13,15,17,27)(14,20,23,25)(16,26,19,29)(18,28,31,43)(21,34,38,39)(22,35,33,40)(24,36,37,41)(30,42,32,44);(14,21)(16,19)(18,32)(20,34)(22,24)(23,38)(25,39)(26,29)(28,44)(30,31)(33,37)(35,36)(40,41)(42,43);(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,22,23,33)(15,26,27,29)(18,30,31,32)(20,35,25,40)(21,37,38,24)(28,42,43,44)(34,41,39,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=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, 401229);