#10040679

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 9947875);