#6672363

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,27,26)(23,28,33,40)(24,29,36,41)(25,30,39,38)(31,42,46,34)(32,37,44,47)(35,43,51,50)(45,52,49,48);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,23)(22,28)(24,34)(25,37)(26,40)(27,33)(29,31)(30,44)(32,38)(35,49)(36,42)(39,47)(41,46)(43,48)(45,51)(50,52);(2,4)(21,24,27,36)(22,29,26,41)(23,31,33,46)(25,35,39,51)(28,42,40,34)(30,43,38,50)(32,45,44,49)(37,52,47,48);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,22,27,26)(23,28,33,40)(24,29,36,41)(25,38,39,30)(31,42,46,34)(32,47,44,37)(35,50,51,43)(45,48,49,52);(7,15)(10,13)(14,20)(18,19)(21,25,27,39)(22,30,26,38)(23,32,33,44)(24,35,36,51)(28,37,40,47)(29,43,41,50)(31,45,46,49)(34,48,42,52)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579559);