#10040647

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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)(3,6)(4,12)(5,7)(8,10)(9,13)(11,15)(14,16)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,34)(35,44)(36,41)(37,48)(38,43)(39,46)(40,47)(42,45);(1,3,5,10)(2,6,7,8)(4,11,13,16)(9,14,12,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35)(34,39)(36,37)(38,46)(40,48)(41,42)(43,44)(45,47);(2,7)(3,10)(9,12)(11,16)(18,20)(23,25)(24,28)(30,31)(33,36,38,47)(34,40,43,41)(35,37,46,45)(39,48,44,42);(1,3)(2,8)(4,11)(5,10)(6,7)(9,15)(12,14)(13,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36)(34,41)(35,45)(37,46)(38,47)(39,48)(40,43)(42,44);(1,2)(3,8)(4,12)(5,7)(6,10)(9,13)(11,14)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,34)(35,44)(36,41)(37,48)(38,43)(39,46)(40,47)(42,45);(1,4,5,13)(2,9,7,12)(3,11,10,16)(6,14,8,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37,38,45)(34,42,43,48)(35,36,46,47)(39,41,44,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=ghi,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, 9947843);