#9923910

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

4151 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,26)(20,25)(21,29)(22,28)(23,30)(24,31)(27,32)(33,34)(35,38)(36,39)(37,45)(40,42)(41,48)(43,44)(46,47);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19)(18,23)(20,29)(21,24)(22,30)(25,32)(26,28)(27,31)(33,35,40,39)(34,38,42,36)(37,43,48,47)(41,46,45,44);(2,4)(3,10)(9,12)(11,16)(17,20,22,31)(18,24,28,25)(19,29,30,27)(21,26,32,23)(33,36,40,38)(34,35,42,39)(37,46,48,44)(41,47,45,43);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20)(18,25)(19,29)(21,26)(22,31)(23,32)(24,28)(27,30)(33,37,40,48)(34,41,42,45)(35,43,39,47)(36,44,38,46);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,19)(18,26)(20,29)(21,25)(22,30)(23,28)(24,32)(27,31)(33,38,40,36)(34,35,42,39)(37,46,48,44)(41,43,45,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21)(18,27)(19,24)(20,26)(22,32)(23,31)(25,30)(28,29)(33,39,40,35)(34,38,42,36)(37,47,48,43)(41,46,45,44)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=i,e^2=gh,f^2=gh,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, 9831106);