#496759

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×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×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,26,22,27,33,41,20,25,31,39,32,40,44,48)(19,24,29,37,30,38,43,47,28,23,34,35,42,36,45,46);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,28)(21,29)(22,30)(24,25)(26,35)(27,36)(31,34)(32,42)(33,43)(37,39)(38,40)(41,46)(44,45)(47,48);(3,8)(5,7)(10,13)(14,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,25)(26,37)(27,38)(31,34)(32,42)(33,43)(35,39)(36,40)(41,47)(44,45)(46,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(19,28)(26,39)(27,40)(29,34)(30,42)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=g,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cei,d^b=di,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, 403955);