#8746069

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3874 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653265);