#8417258

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	#55868	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324454);