#8898530

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

4260 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)(17,18,22,24)(19,25,20,23)(21,26,29,31)(27,32,28,30)(33,34,38,41)(35,39,43,48)(36,44,37,40)(42,47,45,46);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19,22,20)(18,23,24,25)(21,28,29,27)(26,32,31,30)(33,35,38,43)(34,39,41,48)(36,45,37,42)(40,47,44,46);(4,12)(7,14)(10,16)(13,15)(18,24)(23,25)(26,31)(30,32)(33,36,38,37)(34,40,41,44)(35,42,43,45)(39,46,48,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20,22,19)(18,25,24,23)(21,27,29,28)(26,30,31,32)(33,37,38,36)(34,40,41,44)(35,42,43,45)(39,47,48,46);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,21,22,29)(18,26,24,31)(19,27,20,28)(23,30,25,32)(33,35)(34,39)(36,42)(37,45)(38,43)(40,46)(41,48)(44,47);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31)(33,34)(35,39)(36,44)(37,40)(38,41)(42,47)(43,48)(45,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805726);