#8843498

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

4180 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(20,27)(23,25)(28,32)(30,31)(33,34,38,39)(35,40,43,37)(36,44,47,41)(42,46,45,48);(2,7)(4,9)(5,14)(11,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(34,39)(37,40)(41,44)(46,48);(2,7)(3,10)(8,13)(11,16)(18,19)(23,25)(24,26)(30,31)(33,35,38,43)(34,40,39,37)(36,42,47,45)(41,48,44,46);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35)(34,37)(36,45)(38,43)(39,40)(41,48)(42,47)(44,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,38,47)(34,41,39,44)(35,42,43,45)(37,46,40,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,35)(36,46)(38,40)(39,43)(41,42)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=hi,e^2=gh,f^2=h,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, 8750694);