#10247617

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	#47479	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gi,f^2=h,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, 10154813);