#8395031

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

4138 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8302227);