#6382007

order	512
nilpotency class	3
isoclinism	isoclinic class of #61106
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×#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6289203);