#389119

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(5,6,8,7)(9,10,14,18,15,19,33,17)(11,20,23,16,24,36,40,37)(12,25,28,31,29,32,27,13)(21,30,38,39,26,34,35,22);(1,2)(3,4)(9,11,14,23,15,24,33,40)(10,16,17,20,19,37,18,36)(12,26,27,38,29,21,28,35)(13,30,25,39,31,34,32,22);(2,4)(5,7,8,6)(9,12,15,29)(10,13,19,31)(11,21,24,26)(14,27,33,28)(16,34,37,30)(17,25,18,32)(20,22,36,39)(23,35,40,38);(9,13,15,31)(10,12,19,29)(11,22,24,39)(14,32,33,25)(16,35,37,38)(17,28,18,27)(20,21,36,26)(23,34,40,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296315);