#8633826

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	#55048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=gh,f^2=hi,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, 8541022);