#8533437

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3947 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440633);