#360284

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

1929 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)(7,12,20,29)(8,14,23,31)(9,17)(10,15)(11,16)(13,24,32,25)(18,33)(19,26)(21,27,22,28)(30,34)(35,36);(1,2)(3,4)(5,7,10,21,9,20,11,22)(6,12,15,27,17,29,16,28)(8,19,24,18,23,35,25,34)(13,30,14,26,32,33,31,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,16)(15,31)(17,32)(19,21)(20,34)(22,35)(27,30)(28,33)(29,36);(5,8)(6,14)(7,19)(9,23)(10,24)(11,25)(12,26)(13,16)(15,32)(17,31)(18,21)(20,35)(22,34)(27,33)(28,30)(29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=f,f^2=i,g^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, 267480);