#9945509

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	#49107	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,14)(12,16)(17,18,21,22)(19,23,26,30)(20,27,29,24)(25,32,28,31)(33,34,38,36)(35,40,43,39)(37,41,45,44)(42,47,48,46);(2,7)(3,11)(9,16)(10,13)(17,18,21,22)(19,23,26,30)(20,24,29,27)(25,31,28,32)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(2,7)(3,11)(9,16)(10,13)(18,22)(23,30)(24,27)(31,32);(1,3,6,11)(2,8,7,14)(4,10,12,13)(5,9,15,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31)(33,36,38,34)(35,39,43,40)(37,41,45,44)(42,47,48,46);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,20)(18,24)(19,25)(21,29)(22,27)(23,31)(26,28)(30,32)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,e^2=gh,f^2=g,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, 9852705);