#8772853

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

4238 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8680049);