#9935864

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

4268 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9843060);