#8407153

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

4143 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)(3,11)(4,9)(5,10)(6,8)(7,14)(12,16)(13,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34)(35,39)(36,45)(37,44)(38,40)(41,46)(42,48)(43,47);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,13,16,9)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(33,35,38,43)(34,39,40,47)(36,44,46,42)(37,41,48,45);(2,8)(5,16)(7,11)(9,13)(19,28)(22,31)(24,32)(26,29)(34,40)(35,43)(41,45)(42,44);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,11,16,7)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,36)(34,41)(35,44)(37,47)(38,46)(39,48)(40,45)(42,43);(1,3)(2,7)(4,10)(5,13)(6,14)(8,11)(9,16)(12,15)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,37,38,48)(34,42,40,44)(35,45,43,41)(36,39,46,47);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,11,15,7)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,48)(45,46)

Polycyclic presentation:

2^9|b^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8314349);