#9909945

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

4148 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 9817141);