#497687

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=dei,d^c=di

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, 404883);