#8618113

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

4033 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)(6,8)(9,14)(11,15)(13,16)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(35,42)(36,39)(37,46)(40,45);(2,5)(8,10)(9,12)(15,16)(18,23)(20,29)(24,30)(26,32)(33,35,38,42)(34,39,41,36)(37,47,46,43)(40,44,45,48);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,36)(34,35)(37,44)(38,39)(40,43)(41,42)(45,47)(46,48);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,37,38,46)(34,40,41,45)(35,43,42,47)(36,44,39,48);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,34)(35,39)(36,42)(37,40)(38,41)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=g,e^2=gi,f^2=g,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, 8525309);