#6437278

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1252 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)(27,32)(28,39)(29,34)(30,44)(31,35)(33,41)(36,50)(37,46)(38,47)(40,49)(42,53)(43,48)(45,51)(52,55)(54,56);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27)(26,32)(28,36)(29,43)(30,37)(31,38)(33,45)(34,48)(35,47)(39,50)(40,54)(41,51)(42,52)(44,46)(49,56)(53,55);(11,17)(14,20)(18,23)(21,24)(25,28,30,41,31,42,29,40)(26,33,34,39,35,49,44,53)(27,36,37,51,38,52,43,54)(32,45,48,50,47,56,46,55);(10,15)(14,20)(16,22)(21,24)(25,29,31,30)(26,34,35,44)(27,37,38,43)(28,40,42,41)(32,46,47,48)(33,39,49,53)(36,51,52,54)(45,55,56,50);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,38)(32,47)(36,52)(37,43)(45,56)(46,48)(50,55)(51,54)

Polycyclic presentation:

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

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, 6344474);