#8137608

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	#45727	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8044804);