#8814023

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

4136 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,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, 8721219);