#8602613

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

4082 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)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,34,39,42)(35,40,45,38)(36,46,37,41)(43,47,44,48);(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)(24,28)(26,30)(31,32)(33,35,39,45)(34,40,42,38)(36,44,37,43)(41,47,46,48);(2,8)(4,15)(7,9)(10,14)(21,30)(24,32)(26,29)(28,31)(33,36,39,37)(34,41,42,46)(35,43,45,44)(38,47,40,48);(1,3)(2,7)(4,10)(5,16)(6,12)(8,9)(11,13)(14,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,37,39,36)(34,41,42,46)(35,44,45,43)(38,47,40,48);(1,4)(2,5)(3,10)(6,15)(7,16)(8,13)(9,11)(12,14)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,34)(35,38)(36,46)(37,41)(39,42)(40,45)(43,48)(44,47);(1,5)(2,4)(3,11)(6,13)(7,14)(8,15)(9,10)(12,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,38)(34,35)(36,47)(37,48)(39,40)(41,43)(42,45)(44,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,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, 8509809);