#10047334

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,18,23,29)(19,24,28,22)(20,27,31,25)(21,32,30,26)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,46)(47,48);(3,4)(7,10)(11,12)(15,16)(17,19,23,28)(18,24,29,22)(20,21,31,30)(25,26,27,32)(34,40)(36,43)(41,44)(45,47);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,20,23,31)(18,25,29,27)(19,30,28,21)(22,26,24,32)(33,35,39,38)(34,41,40,44)(36,45,43,47)(37,48,42,46);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21,23,30)(18,26,29,32)(19,20,28,31)(22,27,24,25)(33,36,39,43)(34,42,40,37)(35,45,38,47)(41,46,44,48);(1,2)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,20)(18,27)(19,30)(21,28)(22,32)(23,31)(24,26)(25,29)(33,37,39,42)(34,43,40,36)(35,46,38,48)(41,45,44,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22,23,24)(18,28,29,19)(20,26,31,32)(21,27,30,25)(33,38,39,35)(34,44,40,41)(36,47,43,45)(37,46,42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=gi,e^2=g,f^2=gh,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, 9954530);