#6436888

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344084);