#9794340

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

4095 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)(3,9,11,4)(5,8,14,15)(10,13,16,12)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(2,7)(4,9)(8,15)(12,13)(18,24)(19,21)(26,30)(27,29)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4)(2,3)(5,13)(6,9)(7,11)(8,16)(10,15)(12,14)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,5,6,14)(2,8,7,15)(3,10,11,16)(4,12,9,13)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(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=h,d^2=h,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, 9701536);