#6497440

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

Polycyclic presentation:

2^9|a^2=hi,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, 6404636);