#8893539

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

4258 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8800735);