#494791

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 401987);