#9794294

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
prime radical	#51918	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,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, 9701490);