#8114331

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	#53804	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,30,31)(27,35,38,32)(28,39,40,33)(29,42,43,34)(36,44,49,47)(37,45,51,48)(41,46,53,52)(50,55,56,54);(1,2,4,6)(3,5,8,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(26,31)(28,40)(32,35)(34,42)(36,49)(41,53)(45,48)(50,56);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,38)(26,32,31,35)(28,36,40,49)(29,37,43,51)(33,44,39,47)(34,45,42,48)(41,50,53,56)(46,54,52,55);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,47)(37,48)(39,40)(41,46)(42,43)(44,49)(45,51)(50,55)(52,53)(54,56);(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,28,30,40)(26,33,31,39)(27,36,38,49)(29,41,43,53)(32,44,35,47)(34,46,42,52)(37,50,51,56)(45,54,48,55);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,29,30,43)(26,34,31,42)(27,37,38,51)(28,41,40,53)(32,45,35,48)(33,46,39,52)(36,50,49,56)(44,54,47,55)

Polycyclic presentation:

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

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, 8021527);