#8533148

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,e^2=g,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, 8440344);