#8633923

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^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, 8541119);