#497628

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂×#121	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

832 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)(6,10)(11,12)(13,14,16,21,17,22,26,34,18,20,27,32,28,33,38,42)(15,19,24,30,25,31,37,41,23,29,35,39,36,40,43,44);(5,6,9,10)(7,11,8,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,29)(21,30)(22,31)(26,37)(27,35)(28,36)(32,39)(33,40)(34,41)(38,43)(42,44);(2,4)(5,7,9,8)(6,11,10,12)(15,23)(19,29)(24,35)(25,36)(30,39)(31,40)(37,43)(41,44);(5,8,9,7)(6,11,10,12)(14,20)(19,29)(21,32)(22,33)(30,39)(31,40)(34,42)(41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404824);