#6382011

order	512
nilpotency class	3
isoclinism	isoclinic class of #61109
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289207);