#8926641

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=g,f^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, 8833837);