#9778475

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

4454 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9685671);