#8639921

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	#35044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8547117);