#498722

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

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

480 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,21,25,22,26,33,39,20,23,31,34,32,35,44,45)(19,27,29,40,30,41,43,48,28,24,36,37,42,38,46,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(18,23)(19,28)(25,34)(26,35)(29,36)(30,42)(39,45)(43,46);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,19,20,28)(18,24,23,27)(21,29,31,36)(22,30,32,42)(25,37,34,40)(26,38,35,41)(33,43,44,46)(39,47,45,48);(2,6)(5,8)(10,13)(14,16)(19,28)(24,27)(29,36)(30,42)(37,40)(38,41)(43,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405918);