#9835863

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

4250 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18,23,25)(19,27,21,24)(20,30,26,22)(28,29,31,32)(34,38)(36,44)(39,41)(42,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,23,21)(18,24,25,27)(20,28,26,31)(22,29,30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,6)(7,14)(8,11)(15,16)(18,25)(20,26)(24,27)(28,31)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,23,26)(18,22,25,30)(19,28,21,31)(24,29,27,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,19)(18,24,25,27)(20,31,26,28)(22,29,30,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,22)(18,26)(19,29)(20,25)(21,32)(23,30)(24,31)(27,28)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=hi,e^2=h,f^2=gh,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, 9743059);