#8963713

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

4005 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)(35,42,37,36)(38,40,47,48)(43,46,45,44);(1,2,6,4)(3,7,10,11)(5,8,13,12)(9,14,15,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,39,37)(34,36,41,42)(38,43,47,45)(40,44,48,46);(2,4)(5,13)(7,11)(9,15)(17,19,23,20)(18,24,21,25)(22,28,30,27)(26,32,29,31)(33,36,39,42)(34,35,41,37)(38,46,47,44)(40,45,48,43);(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)(35,36)(37,42)(38,40)(39,41)(43,44)(45,46)(47,48);(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)(33,37,39,35)(34,36,41,42)(38,45,47,43)(40,44,48,46);(1,5)(2,8)(3,9)(4,12)(6,13)(7,14)(10,15)(11,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,38,39,47)(34,40,41,48)(35,43,37,45)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=hi,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, 8870909);