#8987021

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

4052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,18,22,24)(19,25,26,20)(21,23,29,30)(27,31,32,28)(33,34,38,41)(35,39,42,36)(37,40,45,47)(43,46,48,44);(1,3,6,9)(2,7,5,11)(4,10,12,13)(8,15,14,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(2,5)(3,9)(4,12)(15,16)(19,26)(20,25)(21,29)(23,30)(35,42)(36,39)(37,45)(40,47);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,13)(11,16)(17,19,22,26)(18,20,24,25)(21,27,29,32)(23,28,30,31)(33,36,38,39)(34,35,41,42)(37,46,45,44)(40,48,47,43);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,12)(14,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,5,6,2)(3,11,9,7)(4,14,12,8)(10,16,13,15)(17,21,22,29)(18,23,24,30)(19,27,26,32)(20,28,25,31)(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=h,d^2=ghi,f^2=gh,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, 8894217);