#8436678

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
prime radical	#40768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343874);