#6477511

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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	#37445	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2965 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,33)(24,38)(25,30)(26,31)(27,32)(28,36)(29,39)(34,48)(35,44)(37,43)(40,46)(41,45)(42,51)(47,50)(49,52);(2,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,37)(22,28,32,44)(24,34,41,51)(25,35,26,36)(29,42,46,48)(30,33,31,43)(38,52,45,47)(39,49,40,50);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,29)(23,34)(25,39)(26,40)(27,41)(28,42)(30,45)(31,38)(32,46)(33,47)(35,49)(36,50)(37,51)(43,52)(44,48);(7,14)(10,18)(13,15)(17,19)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,39,41,40)(28,33,44,43)(29,45,46,38)(34,49,51,50)(42,47,48,52);(6,12)(10,18)(11,20)(17,19)(21,25,27,26)(22,30,32,31)(23,36,37,35)(24,39,41,40)(28,43,44,33)(29,45,46,38)(34,50,51,49)(42,52,48,47)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=i,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^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 6384707);