#6811834

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1640 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:

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

Polycyclic presentation:

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

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, 6719030);