#8634804

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	#46339	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4027 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,23,25)(19,24,29,32)(20,26,27,21)(22,28,31,30)(33,34,39,38)(35,40,42,37)(36,41,45,46)(43,47,48,44);(2,7)(3,11)(4,14)(13,15)(17,19,23,29)(18,24,25,32)(20,28,27,30)(21,22,26,31)(33,35)(34,40)(36,43)(37,38)(39,42)(41,47)(44,46)(45,48);(2,7)(4,14)(8,10)(12,16)(18,25)(19,29)(21,26)(28,30)(35,42)(37,40)(43,48)(44,47);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,20,23,27)(18,26,25,21)(19,30,29,28)(22,32,31,24)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,21,23,26)(18,27,25,20)(19,31,29,22)(24,28,32,30)(33,37)(34,35)(36,44)(38,42)(39,40)(41,43)(45,47)(46,48);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,22,23,31)(18,28,25,30)(19,26,29,21)(20,24,27,32)(33,38,39,34)(35,37,42,40)(36,41,45,46)(43,47,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=h,e^2=gh,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, 8542000);