#8137591

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:

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

Polycyclic presentation:

2^9|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, 8044787);