#6497299

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469983
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,36)(28,33)(29,34)(30,43)(31,35)(32,38)(37,51)(39,46)(40,48)(41,49)(42,50)(44,47)(45,54)(52,55)(53,56);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,32)(28,37)(29,38)(30,44)(31,39)(33,45)(34,46)(35,47)(36,43)(40,52)(41,53)(42,54)(48,51)(49,55)(50,56);(2,6)(5,8)(11,17)(14,21)(18,23)(22,24)(25,28,31,41)(26,33,35,49)(27,37,39,53)(29,42,30,40)(32,45,47,55)(34,50,43,48)(36,51,46,56)(38,54,44,52);(11,17)(14,21)(18,23)(22,24)(25,29,31,30)(26,34,35,43)(27,38,39,44)(28,40,41,42)(32,46,47,36)(33,48,49,50)(37,52,53,54)(45,51,55,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(28,41)(33,49)(37,53)(40,42)(45,55)(48,50)(51,56)(52,54)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=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, 6404495);