#8963695

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=hi,e^2=gi,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, 8870891);