#9909822

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

4326 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,25)(19,26,20,21)(22,24,31,32)(27,30,28,29)(33,34,39,41)(35,38,36,40)(37,42,46,48)(43,45,44,47);(3,9)(7,13)(10,15)(14,16)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,32,30)(33,35,39,36)(34,40,41,38)(37,44,46,43)(42,45,48,47);(2,6)(7,13)(8,11)(14,16)(34,41)(35,36)(42,48)(43,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,32,30)(33,36,39,35)(34,38,41,40)(37,44,46,43)(42,45,48,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21)(18,19)(20,25)(22,30)(23,26)(24,28)(27,32)(29,31)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,32)(26,30)(33,38)(34,35)(36,41)(37,45)(39,40)(42,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,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, 9817018);