#8700562

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	#39525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8607758);