#6677281

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1735 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,25,29)(23,31,35,27)(24,36,37,28)(26,30,39,44)(32,42,34,40)(33,41,48,45)(38,49,50,43)(46,51,47,52);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,27)(24,33)(25,35)(26,32)(28,41)(29,31)(30,40)(34,39)(36,45)(37,48)(38,46)(42,44)(43,51)(47,50)(49,52);(2,4)(23,32)(24,37)(26,39)(27,40)(28,36)(30,44)(31,42)(33,47)(34,35)(41,52)(45,51)(46,48);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,25,37)(22,28,29,36)(23,33,35,48)(26,38,39,50)(27,41,31,45)(30,43,44,49)(32,46,34,47)(40,51,42,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(26,39)(27,42)(30,44)(31,40)(32,35)(33,47)(38,50)(41,52)(43,49)(45,51)(46,48)

Polycyclic presentation:

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

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, 6584477);