#10040674

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	#52839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=hi,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, 9947870);