#8425529

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

4215 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332725);