#9847644

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

4183 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,13)(14,15)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,3,7,6)(2,8,4,10)(5,11,13,14)(9,15,12,16)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(2,4)(8,10)(9,12)(15,16)(18,25)(19,30)(20,27)(28,32)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,4,7,2)(3,10,6,8)(5,12,13,9)(11,16,14,15)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,6,7,3)(2,10,4,8)(5,11,13,14)(9,15,12,16)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=ghi,f^2=h,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, 9754840);