#496972

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 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)(6,9)(7,11)(13,14,16,21)(15,24,25,19)(17,20,28,36)(18,31,29,23)(22,38,37,30)(26,32,41,35)(27,34,42,40)(33,44,39,43);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,15,16,25)(14,19,21,24)(17,26,28,41)(18,27,29,42)(20,35,36,32)(22,33,37,39)(23,34,31,40)(30,44,38,43);(6,9)(10,12)(14,20)(15,25)(17,30)(18,29)(19,32)(21,36)(22,23)(24,35)(26,43)(28,38)(31,37)(33,40)(34,39)(41,44);(5,7,8,11)(6,10,9,12)(14,21)(15,25)(20,36)(22,37)(23,31)(26,41)(27,42)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404168);