#10040730

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947926);