#387956

order	512
nilpotency class	3
isoclinism	isoclinic class of #213752
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1143 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)(3,5)(4,7)(6,8)(9,10,14,18,13,17,24,32)(11,15,21,28,20,27,25,33)(12,16,23,31,22,30,36,40)(19,29,35,39,34,26,37,38);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,25)(16,29)(17,27)(18,33)(21,24)(22,34)(23,37)(26,30)(28,32)(31,38)(35,36)(39,40);(1,4,6,5)(2,7,8,3)(9,12,13,22)(10,16,17,30)(11,19,20,34)(14,23,24,36)(15,26,27,29)(18,31,32,40)(21,35,25,37)(28,38,33,39);(1,5,6,4)(2,7,8,3)(11,20)(14,24)(18,32)(19,34)(23,36)(28,33)(31,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,c^2=eg,d^2=egi,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,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, 295152);