#10040681

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3875 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)(3,7,9,10)(5,12,13,8)(11,16,15,14)(17,18,22,24)(19,25,26,20)(21,29,31,23)(27,28,32,30)(33,34,39,36)(35,43,44,40)(37,41,47,45)(38,42,48,46);(2,4)(3,9)(8,12)(11,15)(19,26)(20,25)(27,32)(28,30)(33,35)(34,40)(36,43)(37,38)(39,44)(41,46)(42,45)(47,48);(2,4)(3,9)(8,12)(11,15)(17,19,22,26)(18,20,24,25)(21,27,31,32)(23,28,29,30)(33,34,39,36)(35,40,44,43)(37,45,47,41)(38,42,48,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,30)(31,32)(33,36,39,34)(35,40,44,43)(37,45,47,41)(38,46,48,42);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,20)(18,19)(21,30)(22,25)(23,32)(24,26)(27,29)(28,31)(33,37,39,47)(34,41,36,45)(35,38,44,48)(40,46,43,42);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21,22,31)(18,23,24,29)(19,27,26,32)(20,28,25,30)(33,38)(34,42)(35,37)(36,46)(39,48)(40,45)(41,43)(44,47)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,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, 9947877);