#10040795

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gi,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, 9947991);