#10040445

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3902 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,29)(19,30,31,24)(20,26,32,25)(21,27,28,22)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,3,6,11)(2,7,8,16)(4,14,10,12)(5,13,9,15)(17,19)(18,24)(20,22)(21,25)(23,31)(26,28)(27,32)(29,30)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(2,8)(3,11)(4,10)(13,15)(17,20,23,32)(18,25,29,26)(19,22,31,27)(21,30,28,24)(33,35,37,42)(34,38,40,46)(36,41,44,43)(39,45,48,47);(1,4,6,10)(2,9,8,5)(3,12,11,14)(7,13,16,15)(17,20)(18,26)(19,27)(21,24)(22,31)(23,32)(25,29)(28,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,3)(2,7)(4,12)(5,15)(6,11)(8,16)(9,13)(10,14)(17,21)(18,27)(19,26)(20,24)(22,29)(23,28)(25,31)(30,32)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(1,5,6,9)(2,10,8,4)(3,13,11,15)(7,12,16,14)(17,22,23,27)(18,28,29,21)(19,20,31,32)(24,26,30,25)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=i,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, 9947641);