#8634834

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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:

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

Polycyclic presentation:

2^9|a^2=hi,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, 8542030);