#8957583

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

4198 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8864779);