#10040835

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3811 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)(17,18,23,26)(19,24,22,32)(20,25,30,21)(27,28,29,31)(35,38)(36,40)(43,46)(44,48);(2,4)(3,9)(8,12)(11,15)(17,19,23,22)(18,24,26,32)(20,27,30,29)(21,31,25,28)(33,34,39,42)(35,36,38,40)(37,45,47,41)(43,48,46,44);(2,4)(3,9)(8,12)(11,15)(17,20,23,30)(18,21,26,25)(19,27,22,29)(24,31,32,28)(33,35,39,38)(34,40,42,36)(37,46,47,43)(41,44,45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,25)(19,27)(21,26)(22,29)(23,30)(24,28)(31,32)(33,36,39,40)(34,35,42,38)(37,44,47,48)(41,43,45,46);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,21,23,25)(18,20,26,30)(19,28,22,31)(24,29,32,27)(33,37,39,47)(34,41,42,45)(35,43,38,46)(36,44,40,48);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,22,23,19)(18,24,26,32)(20,29,30,27)(21,31,25,28)(33,38,39,35)(34,40,42,36)(37,43,47,46)(41,44,45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=i,e^2=h,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, 9948031);