#9625449

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

(9,10,15,12)(11,18,14,16)(13,20,22,17)(19,23,21,24)(25,26)(27,36)(28,29)(30,45)(31,32)(33,37)(34,43)(35,46)(38,39)(40,47)(41,48)(42,49)(44,55)(50,51)(52,56)(53,54);(1,2)(3,4)(5,6)(7,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(26,32)(27,37)(29,34)(35,45)(38,41)(40,51)(44,49)(52,54);(2,6)(3,7)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,27)(26,33)(28,41)(29,39)(30,40)(31,37)(32,36)(34,48)(35,47)(38,43)(42,54)(44,53)(45,50)(46,51)(49,56)(52,55);(3,7)(4,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,28)(26,34)(27,38)(29,32)(30,42)(31,43)(33,39)(35,49)(36,48)(37,41)(40,52)(44,45)(46,55)(47,53)(50,56)(51,54);(1,3)(2,4)(5,7)(6,8)(25,29,31,34)(26,28,32,43)(27,39,37,48)(30,44,46,49)(33,41,36,38)(35,42,45,55)(40,53,51,56)(47,54,50,52);(1,4)(2,3)(5,8)(6,7)(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(25,30,31,46)(26,35,32,45)(27,40,37,51)(28,42,43,55)(29,44,34,49)(33,47,36,50)(38,52,41,54)(39,53,48,56)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532645);