#8926649

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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, 8833845);