#8348929

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

4171 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=g,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, 8256125);