#8588360

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

4241 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)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,34)(35,43)(36,45)(37,44)(38,42)(39,40)(41,47)(46,48);(1,3,7,9)(2,4,5,12)(6,10,15,16)(8,11,13,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32)(33,35,39,44)(34,37,40,43)(36,46,47,38)(41,48,45,42);(2,5)(3,9)(8,13)(10,16)(21,30)(24,32)(26,29)(28,31)(34,40)(37,43)(41,45)(42,48);(1,4)(2,3)(5,9)(6,14)(7,12)(8,16)(10,13)(11,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,36,39,47)(34,41,40,45)(35,38,44,46)(37,42,43,48);(1,5,7,2)(3,4,9,12)(6,13,15,8)(10,11,16,14)(17,21,22,30)(18,24,25,32)(19,26,27,29)(20,28,23,31)(33,37)(34,35)(36,42)(38,41)(39,43)(40,44)(45,46)(47,48);(1,6)(2,8)(3,10)(4,11)(5,13)(7,15)(9,16)(12,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,38)(34,42)(35,36)(37,41)(39,46)(40,48)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,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, 8495556);