#498708

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#2388	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

642 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)(11,13)(12,14)(15,16)(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)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(25,34)(26,35)(29,36)(30,42)(39,45)(43,46);(3,8)(5,7)(10,14)(11,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,37)(26,38)(31,36)(32,42)(33,43)(34,40)(35,41)(39,47)(44,46)(45,48);(2,6)(3,8)(10,14)(13,16)(19,28)(24,27)(29,36)(30,42)(37,40)(38,41)(43,46)(47,48)

Polycyclic presentation:

2^9|a^2=g,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, 405904);