#9863087

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,20)(19,21,24,26)(22,25,31,28)(27,29,32,30)(33,34,38,42)(35,39,37,41)(36,40,45,44)(43,47,46,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(19,24)(21,26)(27,32)(29,30)(33,34)(35,39)(36,44)(37,41)(38,42)(40,45)(43,48)(46,47);(2,3)(7,9)(8,10)(14,15)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,35,38,37)(34,39,42,41)(36,43,45,46)(40,47,44,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,48)(42,44);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37,38,35)(34,41,42,39)(36,43,45,46)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=i,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770283);