#8678963

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4027 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8586159);