#6496501

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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, 6403697);