#6673656

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 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)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,26)(23,28,34,33)(24,29,37,40)(25,30,38,39)(31,41,45,35)(32,42,46,47)(36,43,49,50)(44,51,52,48);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,28)(24,35)(25,32)(26,33)(27,34)(29,31)(30,42)(36,48)(37,41)(38,46)(39,47)(40,45)(43,44)(49,51)(50,52);(2,4)(21,24,27,37)(22,29,26,40)(23,31,34,45)(25,36,38,49)(28,41,33,35)(30,43,39,50)(32,44,46,52)(42,51,47,48);(6,11)(10,13)(12,19)(18,20)(21,25)(22,30)(23,32)(24,36)(26,39)(27,38)(28,42)(29,43)(31,44)(33,47)(34,46)(35,48)(37,49)(40,50)(41,51)(45,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,26)(23,33)(25,38)(28,34)(29,40)(31,35)(32,42)(36,49)(41,45)(44,51)(46,47)(48,52)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,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, 6580852);