#9717693

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

4251 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9624889);