#9723150

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

4133 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)(19,23,28,32)(20,29,30,24)(21,25,27,31)(33,34,39,40)(35,43,38,41)(36,37,42,47)(44,45,46,48);(1,3)(2,8)(4,5)(6,10)(7,13)(9,12)(11,15)(14,16)(17,19,22,28)(18,23,26,32)(20,21,30,27)(24,31,29,25)(34,40)(37,47)(41,43)(45,48);(2,9)(3,13)(4,11)(10,16)(17,20,22,30)(18,24,26,29)(19,27,28,21)(23,25,32,31)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,47,48);(1,4,7,11)(2,6,9,14)(3,5,13,15)(8,10,12,16)(17,20)(18,24)(19,21)(22,30)(23,31)(25,32)(26,29)(27,28)(33,36,39,42)(34,37,40,47)(35,44,38,46)(41,48,43,45);(1,5)(2,10)(3,11)(4,13)(6,12)(7,15)(8,14)(9,16)(17,21)(18,25)(19,20)(22,27)(23,29)(24,32)(26,31)(28,30)(33,37)(34,42)(35,45)(36,40)(38,48)(39,47)(41,44)(43,46);(1,6)(2,11)(3,10)(4,9)(5,12)(7,14)(8,15)(13,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,27)(28,32)(29,30)(33,38,39,35)(34,43,40,41)(36,46,42,44)(37,45,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,f^2=hi,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, 9630346);