#8814021

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

4153 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)(3,5)(4,9)(6,7)(8,10)(11,13)(12,14)(15,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34,38,39)(35,40,43,37)(36,41,46,47)(42,48,44,45);(2,7)(4,12)(5,8)(11,16)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(34,39)(37,40)(41,47)(45,48);(2,7)(3,10)(4,12)(13,15)(18,19)(20,27)(24,26)(28,32)(33,35,38,43)(34,40,39,37)(36,44,46,42)(41,45,47,48);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,4,6,12)(2,9,7,14)(3,11,10,16)(5,13,8,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,41)(35,42)(37,45)(38,46)(39,47)(40,48)(43,44);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,35)(36,45)(38,40)(39,43)(41,42)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721217);