#386412

order	512
nilpotency class	3
isoclinism	isoclinic class of #31414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

955 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=efg,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=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, 293608);