#8617803

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	#39500	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,e^2=ghi,f^2=gh,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, 8524999);