#8825894

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

4136 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8733090);