#8630787

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	#40349	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=g,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, 8537983);