#8633314

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	#46333	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8540510);