#6528965

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6436161);