#9980345

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(18,22)(20,28)(23,25)(26,32)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,45,47,44);(4,11)(8,14)(10,15)(12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,47,45);(2,7)(3,9)(8,14)(10,15)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,35,38,43)(34,39,41,36)(37,42,46,47)(40,45,48,44);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36)(34,35)(37,45)(38,39)(40,47)(41,43)(42,48)(44,46);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,45)(41,48)(43,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9887541);