#8757574

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8664770);