#8963639

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8870835);