#6673785

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1705 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580981);