#387416

order	512
nilpotency class	3
isoclinism	isoclinic class of #213760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1198 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,f^d=fi,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, 294612);