#10247044

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	#55360	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3848 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154240);