#9720533

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

4287 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)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,29)(30,31)(33,34)(35,41)(36,42)(37,47)(38,39)(40,45)(43,48)(44,46);(4,11)(8,14)(10,15)(12,16)(17,19)(18,21)(20,28)(22,26)(23,25)(24,32)(27,30)(29,31)(34,39)(36,46)(40,41)(43,47);(2,7)(3,9)(8,14)(10,15)(18,23)(19,26)(24,31)(27,28)(33,35,38,45)(34,40,39,41)(36,43,46,47)(37,44,48,42);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,30)(31,32)(33,35)(34,41)(36,43)(37,44)(38,45)(39,40)(42,48)(46,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,22,30)(18,24,23,31)(19,27,26,28)(21,29,25,32)(33,36,38,46)(34,42,39,44)(35,43,45,47)(37,41,48,40);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,31)(30,32)(33,37,38,48)(34,43,39,47)(35,44,45,42)(36,40,46,41)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,e^2=g,f^2=gi,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, 9627729);