#391845

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

961 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:

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

Polycyclic presentation:

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

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, 299041);