#9724819

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

3941 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)(4,7)(5,10)(6,11)(8,12)(9,16)(14,15)(17,18)(19,27)(20,24)(21,31)(22,26)(23,28)(25,29)(30,32)(33,34)(35,38)(36,42)(37,47)(39,41)(40,43)(44,48)(45,46);(1,3)(2,9)(4,6)(5,14)(7,15)(8,16)(10,11)(12,13)(17,19,22,28)(18,23,26,27)(20,29,30,21)(24,31,32,25)(33,35,40,37)(34,38,43,47)(36,45,39,44)(41,48,42,46);(1,4,8,5)(2,7,12,10)(3,6,16,14)(9,15,13,11)(17,18,22,26)(19,23,28,27)(20,24,30,32)(21,25,29,31)(33,36,40,39)(34,41,43,42)(35,44,37,45)(38,46,47,48);(1,5,8,4)(2,10,12,7)(3,14,16,6)(9,11,13,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,31)(30,32)(33,37,40,35)(34,38,43,47)(36,44,39,45)(41,48,42,46);(1,6,8,14)(2,11,12,15)(3,4,16,5)(7,9,10,13)(17,20,22,30)(18,24,26,32)(19,21,28,29)(23,25,27,31)(33,38)(34,35)(36,46)(37,43)(39,48)(40,47)(41,44)(42,45);(1,7,8,10)(2,4,12,5)(3,15,16,11)(6,13,14,9)(17,21)(18,25)(19,20)(22,29)(23,24)(26,31)(27,32)(28,30)(33,39,40,36)(34,42,43,41)(35,44,37,45)(38,46,47,48)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=gh,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, 9632015);