#8926567

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35463	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833763);