#9923658

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

4254 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9830854);