#6386264

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2836 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)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,28)(24,35)(25,30)(26,39)(27,31)(29,36)(32,47)(33,42)(34,43)(37,45)(38,44)(40,46)(41,50)(48,52)(49,51);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23)(22,28)(24,32)(25,38)(26,34)(27,33)(29,41)(30,44)(31,42)(35,47)(36,50)(37,48)(39,43)(40,49)(45,52)(46,51);(21,24)(22,29)(23,32)(25,36)(26,40)(27,37)(28,41)(30,45)(31,46)(33,48)(34,49)(35,39)(38,50)(42,51)(43,47)(44,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,33)(28,42)(32,48)(34,38)(41,51)(43,44)(47,52)(49,50);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,31,39)(23,34,33,38)(24,36,37,40)(28,43,42,44)(29,45,46,35)(32,49,48,50)(41,47,51,52)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293460);