#8348917

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

4247 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,27)(19,24,28,32)(20,21,25,30)(22,26,31,29)(33,34,38,39)(35,42,40,36)(37,41,46,47)(43,48,45,44);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,19)(18,24)(20,22)(21,26)(23,28)(25,31)(27,32)(29,30)(34,39)(36,42)(41,47)(44,48);(4,12)(7,14)(10,16)(13,15)(19,28)(22,31)(24,32)(26,29)(33,35,38,40)(34,36,39,42)(37,45,46,43)(41,48,47,44);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,31)(29,32)(33,34)(35,36)(37,41)(38,39)(40,42)(43,44)(45,48)(46,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,21)(18,25)(19,29)(20,27)(22,24)(23,30)(26,28)(31,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,45)(43,47)(46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22)(18,26)(19,20)(21,24)(23,31)(25,28)(27,29)(30,32)(33,37,38,46)(34,41,39,47)(35,43,40,45)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8256113);