#8533564

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	#58069	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440760);