#8276612

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

4311 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8183808);