#10047208

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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)(17,18,22,27)(19,25,28,23)(20,29,30,24)(21,31,32,26)(33,34)(35,37)(36,42)(38,41)(39,43)(40,44)(45,47)(46,48);(3,4)(7,10)(11,12)(15,16)(19,28)(20,30)(23,25)(24,29)(33,35,40,38)(34,41,44,37)(36,45,39,46)(42,48,43,47);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,22,28)(18,23,27,25)(20,31,30,26)(21,24,32,29)(33,36,40,39)(34,42,44,43)(35,45,38,46)(37,47,41,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,22,30)(18,24,27,29)(19,26,28,31)(21,25,32,23)(33,37,40,41)(34,35,44,38)(36,47,39,48)(42,45,43,46);(1,2)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,19)(18,25)(20,26)(21,29)(22,28)(23,27)(24,32)(30,31)(33,38,40,35)(34,41,44,37)(36,45,39,46)(42,47,43,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21,22,32)(18,26,27,31)(19,24,28,29)(20,25,30,23)(33,39,40,36)(34,43,44,42)(35,46,38,45)(37,48,41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,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, 9954404);