#8755346

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

4119 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662542);