#9724822

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

3939 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32)(36,44)(40,47)(43,45)(46,48);(1,3)(2,8)(4,6)(5,14)(7,15)(9,16)(10,11)(12,13)(17,19,22,21)(18,23,26,25)(20,28,29,27)(24,31,32,30)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(1,4,7,5)(2,9,12,10)(3,6,15,14)(8,16,13,11)(17,18,22,26)(19,23,21,25)(20,24,29,32)(27,30,28,31)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,47,48);(1,5,7,4)(2,10,12,9)(3,14,15,6)(8,11,13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,6,7,14)(2,11,12,16)(3,4,15,5)(8,10,13,9)(17,20,22,29)(18,24,26,32)(19,27,21,28)(23,30,25,31)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,2)(3,8)(4,9)(5,10)(6,16)(7,12)(11,14)(13,15)(17,21,22,19)(18,25,26,23)(20,27,29,28)(24,30,32,31)(33,37,38,35)(34,39,41,42)(36,45,44,43)(40,46,47,48)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^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, 9632018);