#9923911

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

4149 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 9831107);