#9920343

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

4229 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)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(33,34)(35,40)(36,46)(37,42)(38,39)(41,47)(43,48)(44,45);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(33,35)(34,40)(36,37)(38,44)(39,45)(41,48)(42,46)(43,47);(2,7)(3,10)(8,15)(11,12)(18,23)(22,29)(24,27)(30,31)(33,36,39,47)(34,41,38,46)(35,43,45,37)(40,42,44,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,36)(34,41)(35,37)(38,46)(39,47)(40,48)(42,44)(43,45);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37)(34,42)(35,36)(38,48)(39,43)(40,46)(41,44)(45,47);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,20,21,28)(18,24,23,27)(19,25,26,32)(22,30,29,31)(33,38,39,34)(35,44,45,40)(36,46,47,41)(37,42,43,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,f^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, 9827539);