#391163

order	512
nilpotency class	3
isoclinism	isoclinic class of #232169
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
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

718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=fh,d^2=efg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,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, 298359);