#8986999

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

4220 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)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,34,38,41)(35,39,42,36)(37,40,45,47)(43,46,48,44);(2,7)(4,15)(8,11)(10,13)(17,19,22,25)(18,23,21,27)(20,26,28,29)(24,31,30,32)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(2,7)(3,12)(4,15)(5,16)(18,21)(19,25)(20,28)(31,32)(35,42)(36,39)(37,45)(40,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,29)(27,32)(33,36,38,39)(34,35,41,42)(37,46,45,44)(40,48,47,43);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,18)(19,27,25,23)(20,30,28,24)(26,32,29,31)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8894195);