#8962976

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

4222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8870172);