#8957601

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

4202 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=hi,e^2=h,f^2=gh,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, 8864797);