#8731167

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	#59687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8638363);