#9938155

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

4153 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)(3,8,10,5)(4,9,12,15)(11,13,16,14)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,39,41)(35,44,46,40)(36,37,42,45)(38,47,48,43);(2,7)(3,10)(4,12)(13,14)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,39,46)(34,40,41,44)(36,47,42,43)(37,38,45,48);(18,19)(23,25)(24,26)(30,31)(34,41)(36,42)(40,44)(43,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,13)(12,16)(14,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,39,42)(34,37,41,45)(35,43,46,47)(38,40,48,44);(1,4)(2,9)(3,11)(5,14)(6,12)(7,15)(8,13)(10,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37)(34,42)(35,38)(36,41)(39,45)(40,43)(44,47)(46,48);(1,5,6,8)(2,3,7,10)(4,13,12,14)(9,16,15,11)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38)(34,43)(35,45)(36,40)(37,46)(39,48)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=h,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, 9845351);