#6672321

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

1682 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);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,22,26,30)(23,31,34,45)(24,35,36,46)(25,38,41,29)(27,39,43,33)(28,40,44,37)(32,42,48,51)(47,50,52,49);(2,4)(21,23,26,34)(22,27,30,43)(24,32,36,48)(25,39,41,33)(28,42,44,51)(29,45,38,31)(35,47,46,52)(37,50,40,49);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,36)(28,44)(32,48)(35,46)(37,40)(42,51)(47,52)(49,50);(7,15)(10,13)(14,20)(18,19)(21,24,26,36)(22,28,30,44)(23,32,34,48)(25,40,41,37)(27,42,43,51)(29,46,38,35)(31,47,45,52)(33,50,39,49)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6579517);