#8603523

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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510719);