#9923898

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

4011 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:

(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34)(35,38)(36,39)(37,45)(40,42)(41,48)(43,44)(46,47);(2,4)(7,9)(8,12)(14,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35,40,39)(34,38,42,36)(37,43,48,47)(41,46,45,44);(18,20)(24,26)(25,28)(31,32)(33,36,40,38)(34,35,42,39)(37,46,48,44)(41,47,45,43);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,37,40,48)(34,41,42,45)(35,43,39,47)(36,44,38,46);(1,4,6,2)(3,9,11,7)(5,8,13,12)(10,14,16,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,38,40,36)(34,35,42,39)(37,46,48,44)(41,43,45,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,19)(18,26,20,24)(21,27,29,30)(25,31,28,32)(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=h,c^2=i,d^2=gi,e^2=gh,f^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, 9831094);