#360286

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1965 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267482);