#8997139

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

4062 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)(3,7,9,11)(5,12,13,8)(10,16,15,14)(34,39)(35,43)(36,46)(42,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18,22,26)(19,24,29,23)(20,21,25,32)(27,31,30,28)(33,34)(35,40)(36,41)(37,47)(38,39)(42,44)(43,45)(46,48);(2,4)(3,9)(5,13)(14,16)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,31,32,28)(33,35,38,43)(34,40,39,45)(36,37,46,44)(41,47,48,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,22,25)(18,21,26,32)(19,27,29,30)(23,28,24,31)(33,36,38,46)(34,41,39,48)(35,44,43,37)(40,42,45,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,37,38,44)(34,42,39,47)(35,36,43,46)(40,48,45,41);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,21)(18,25)(19,28)(20,26)(22,32)(23,30)(24,27)(29,31)(33,34)(35,45)(36,41)(37,42)(38,39)(40,43)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8904335);