#8602614

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^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, 8509810);