#9947520

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

4031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=ghi,e^2=h,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, 9854716);