#8137592

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

(1,2,6,10)(3,11,13,7)(4,14,12,8)(5,9,15,16)(17,18,23,28)(19,29,30,24)(20,26,21,25)(22,31,32,27)(35,42)(37,45)(39,40)(41,44);(1,3,6,13)(2,7,10,11)(4,5,12,15)(8,9,14,16)(17,19,23,30)(18,24,28,29)(20,27,21,31)(22,26,32,25)(33,34)(35,40)(36,38)(37,41)(39,42)(43,47)(44,45)(46,48);(17,20,23,21)(18,25,28,26)(19,27,30,31)(22,29,32,24)(33,35)(34,39)(36,40)(37,43)(38,42)(41,46)(44,47)(45,48);(1,4)(2,8)(3,5)(6,12)(7,9)(10,14)(11,16)(13,15)(17,21,23,20)(18,26,28,25)(19,27,30,31)(22,29,32,24)(33,36,38,34)(35,40,42,39)(37,44,45,41)(43,47,48,46);(1,5)(2,9)(3,12)(4,13)(6,15)(7,14)(8,11)(10,16)(17,22)(18,27)(19,26)(20,29)(21,24)(23,32)(25,30)(28,31)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(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,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, 8044788);