#386527

order	512
nilpotency class	3
isoclinism	isoclinic class of #219532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1044 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efg,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=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, 293723);