#9835889

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

4123 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)(17,18,22,24)(19,26,30,23)(20,29,31,25)(21,28,32,27)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,32)(27,31)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(2,6)(7,14)(8,11)(15,16)(18,24)(20,31)(23,26)(27,28)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20)(18,25)(19,28)(21,26)(22,31)(23,32)(24,29)(27,30)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,19)(18,26)(20,28)(21,25)(22,30)(23,24)(27,31)(29,32)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,21)(18,27)(19,29)(20,26)(22,32)(23,31)(24,28)(25,30)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gh,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, 9743085);