#10040488

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	#40686	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18,22,27)(19,25,28,23)(20,29,30,24)(21,31,32,26)(33,34)(35,40)(36,41)(37,42)(38,39)(43,44)(45,47)(46,48);(3,9)(6,13)(10,15)(14,16)(19,28)(20,30)(23,25)(24,29)(34,39)(36,42)(40,47)(43,48);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,22,28)(18,23,27,25)(20,31,30,26)(21,24,32,29)(33,35)(34,40)(36,43)(37,46)(38,45)(39,47)(41,44)(42,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20,22,30)(18,24,27,29)(19,26,28,31)(21,25,32,23)(33,36,38,42)(34,41,39,37)(35,43,45,48)(40,44,47,46);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,25)(20,26)(21,29)(22,28)(23,27)(24,32)(30,31)(33,37,38,41)(34,42,39,36)(35,44,45,46)(40,43,47,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,21,22,32)(18,26,27,31)(19,24,28,29)(20,25,30,23)(33,35)(34,40)(36,43)(37,46)(38,45)(39,47)(41,44)(42,48)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gi,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, 9947684);