#6673632

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	C₂×#3347	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,27)(23,24,29,28)(26,30,34,36)(31,33,35,32);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(22,27)(23,31)(24,32)(26,34)(28,33)(29,35);(2,4)(21,23,25,29)(22,28,27,24)(26,35,34,31)(30,33,36,32);(6,11)(10,13)(12,19)(18,20)(22,27)(24,28)(30,36)(32,33);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,24,25,28)(22,29,27,23)(26,32,34,33)(30,31,36,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=f,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, 6580828);