#8993172

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	#41930	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=i,e^2=g,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, 8900368);