#8720597

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

4182 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18,23,27)(19,20,30,24)(21,31,29,25)(22,26,28,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,48)(45,46);(3,10)(6,7)(11,16)(14,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,32)(26,31)(33,35,39,37)(34,40,43,41)(36,44,47,46)(38,45,42,48);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,20,23,24)(18,19,27,30)(21,26,29,32)(22,25,28,31)(33,35,39,37)(34,41,43,40)(36,46,47,44)(38,45,42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,19)(21,26)(22,31)(23,24)(25,28)(27,30)(29,32)(33,36,39,47)(34,42,43,38)(35,44,37,46)(40,48,41,45);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,21)(18,25)(19,28)(20,26)(22,30)(23,29)(24,32)(27,31)(33,37,39,35)(34,40,43,41)(36,46,47,44)(38,45,42,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,23,28)(18,26,27,32)(19,29,30,21)(20,25,24,31)(33,38,39,42)(34,36,43,47)(35,45,37,48)(40,44,41,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=ghi,e^2=gh,f^2=g,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, 8627793);