#6497439

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,37)(28,42)(29,34)(30,47)(31,36)(32,40)(33,44)(35,38)(39,51)(41,50)(43,56)(45,48)(46,53)(49,54)(52,55);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,38,54,31,41,30,40)(26,32,47,50,36,49,35,37)(28,29,39,55,43,45,44,46)(33,48,56,52,51,34,42,53);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,28,30,44,31,43,38,39)(26,33,35,42,36,51,47,56)(27,29,40,46,41,45,54,55)(32,48,37,53,49,34,50,52);(11,17)(14,21)(18,23)(22,24)(25,29)(26,34)(27,39)(28,40)(30,46)(31,45)(32,42)(33,50)(35,52)(36,48)(37,51)(38,55)(41,44)(43,54)(47,53)(49,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(28,43)(29,45)(33,51)(34,48)(39,44)(42,56)(46,55)(52,53)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=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, 6404635);