#6498150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
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

1358 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,35)(28,33)(29,41)(30,32)(31,37)(34,43)(36,49)(38,48)(39,46)(40,53)(42,44)(45,51)(47,52)(50,56)(54,55);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,38)(26,31,32,44)(28,36,39,50)(29,42,43,37)(33,45,46,55)(34,48,41,35)(40,54,52,51)(47,56,53,49);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(26,32)(31,44)(33,46)(34,41)(35,48)(45,55)(47,53)(49,56);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,28)(26,33)(27,36)(29,40)(30,39)(31,45)(32,46)(34,47)(35,49)(37,51)(38,50)(41,53)(42,54)(43,52)(44,55)(48,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(28,39)(33,46)(36,50)(40,52)(45,55)(47,53)(49,56)(51,54)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=gi,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,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, 6405346);