#9912943

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	#37655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4559 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9820139);