#9947489

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	#48752	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(17,18,24,26)(19,20,21,22)(23,31,32,25)(27,30,29,28)(35,41)(38,42)(43,47)(46,48);(2,7)(3,11)(9,16)(10,13)(17,19)(18,20)(21,24)(22,26)(23,27)(25,28)(29,32)(30,31)(33,34,37,40)(35,42,41,38)(36,39,45,44)(43,48,47,46);(2,7)(3,11)(9,16)(10,13)(17,20,24,22)(18,19,26,21)(23,28,32,30)(25,27,31,29)(33,34)(35,38)(36,39)(37,40)(41,42)(43,46)(44,45)(47,48);(1,3,6,11)(2,8,7,14)(4,10,12,13)(5,9,15,16)(17,21,24,19)(18,20,26,22)(23,29,32,27)(25,28,31,30)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,22,24,20)(18,19,26,21)(23,28,32,30)(25,29,31,27)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,23)(18,25)(19,27)(20,28)(21,29)(22,30)(24,32)(26,31)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gi,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, 9854685);