#9778479

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

4292 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)(3,12)(4,14)(5,9)(6,13)(7,8)(10,15)(11,16)(17,18)(19,23)(20,25)(21,22)(24,30)(26,32)(27,28)(29,31);(2,8)(3,5)(10,14)(11,13)(17,19,22,28)(18,23,21,27)(20,29,30,26)(24,32,25,31)(33,34,39,42)(35,36,37,40)(38,41,46,48)(43,44,45,47);(1,3,7,5)(2,9,8,12)(4,11,15,13)(6,10,16,14)(17,20)(18,24)(19,26)(21,25)(22,30)(23,31)(27,32)(28,29)(33,35,39,37)(34,40,42,36)(38,43,46,45)(41,47,48,44);(1,4,7,15)(2,10,8,14)(3,13,5,11)(6,12,16,9)(17,19)(18,23)(20,26)(21,27)(22,28)(24,31)(25,32)(29,30)(33,36,39,40)(34,35,42,37)(38,44,46,47)(41,43,48,45);(1,5,7,3)(2,9,8,12)(4,11,15,13)(6,14,16,10)(17,20)(18,25)(19,26)(21,24)(22,30)(23,32)(27,31)(28,29)(33,37,39,35)(34,40,42,36)(38,43,46,45)(41,44,48,47);(1,6)(2,11)(3,10)(4,12)(5,14)(7,16)(8,13)(9,15)(17,21,22,18)(19,27,28,23)(20,25,30,24)(26,32,29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9685675);