#8718300

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

4210 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,23,27)(19,24,28,31)(20,25,29,30)(21,22,26,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19,23,28)(18,24,27,31)(20,21,29,26)(22,30,32,25)(33,35)(34,38)(36,42)(37,43)(39,41)(40,47)(44,48)(45,46);(4,13)(8,12)(11,14)(15,16)(19,28)(20,29)(24,31)(25,30)(33,34,37,41)(35,39,43,38)(36,40,46,44)(42,48,45,47);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,20,23,29)(18,25,27,30)(19,21,28,26)(22,31,32,24)(33,34)(35,39)(36,44)(37,41)(38,43)(40,46)(42,47)(45,48);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,21)(18,22)(19,29)(20,28)(23,26)(24,30)(25,31)(27,32)(33,35)(34,39)(36,45)(37,43)(38,41)(40,47)(42,46)(44,48);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,22)(18,26)(19,30)(20,24)(21,27)(23,32)(25,28)(29,31)(33,36,37,46)(34,40,41,44)(35,42,43,45)(38,47,39,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625496);