#9057446

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

4052 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:

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

Polycyclic presentation:

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

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, 8964642);