#9850587

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

4041 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,22,26)(19,21,25,23)(20,24,30,32)(27,31,28,29)(34,38)(36,44)(39,41)(42,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,22,25)(18,23,26,21)(20,28,30,27)(24,31,32,29)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,7)(6,13)(8,11)(14,15)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20,22,30)(18,24,26,32)(19,27,25,28)(21,31,23,29)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,32,28)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9757783);