#9923673

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

4364 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)(3,9)(4,5)(6,8)(7,13)(10,15)(11,14)(12,16)(17,18)(19,24)(20,29)(21,26)(22,30)(23,25)(27,31)(28,32)(34,38)(36,44)(39,41)(42,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,9)(10,16)(14,15)(17,19,23,27)(18,24,25,31)(20,28,30,21)(22,26,29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,8)(4,10)(7,9)(11,16)(18,25)(19,27)(20,30)(26,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,4,6,10)(2,5,8,15)(3,11,13,16)(7,12,9,14)(17,20)(18,22)(19,28)(21,27)(23,30)(24,26)(25,29)(31,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,9)(4,11)(5,14)(6,13)(7,8)(10,16)(12,15)(17,21,23,28)(18,26,25,32)(19,20,27,30)(22,24,29,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5)(2,10)(3,12)(4,8)(6,15)(7,16)(9,11)(13,14)(17,22)(18,20)(19,26)(21,31)(23,29)(24,28)(25,30)(27,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830869);