#8837412

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

4125 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)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18)(19,25)(20,22)(21,28)(23,27)(24,30)(26,31)(29,32)(33,34,38,42)(35,43,44,39)(36,40,46,48)(37,45,47,41);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(35,44)(37,47)(39,43)(41,45);(3,11)(4,13)(7,12)(8,16)(18,20)(23,25)(24,28)(31,32)(33,35)(34,39)(36,45)(37,40)(38,44)(41,46)(42,43)(47,48);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,36)(34,40)(35,41)(37,43)(38,46)(39,47)(42,48)(44,45);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,35)(34,39)(36,45)(37,40)(38,44)(41,46)(42,43)(47,48);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,37)(34,41)(35,40)(36,43)(38,47)(39,46)(42,45)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=hi,e^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, 8744608);