#391381

order	512
nilpotency class	3
isoclinism	isoclinic class of #219525
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
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

725 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298577);