#8639787

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	#39785	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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)(19,21)(20,24)(22,23)(25,26)(27,31)(28,29)(30,32)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(1,2)(3,9)(4,7)(5,8)(6,10)(11,15)(12,13)(14,16)(17,19,22,26)(18,21,23,25)(20,28,30,27)(24,29,32,31)(35,41)(39,43)(42,47)(46,48);(3,10)(6,9)(11,16)(14,15)(18,23)(19,26)(24,32)(27,28)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,15)(14,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,32)(25,29)(26,28)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,3,5,10)(2,6,8,9)(4,11,13,16)(7,14,12,15)(17,19)(18,25)(20,27)(21,23)(22,26)(24,29)(28,30)(31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,21,22,25)(18,19,23,26)(20,29,30,31)(24,28,32,27)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,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, 8546983);