#215265

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1036 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)(3,7)(4,6)(5,8)(9,10)(11,17,18,15)(12,20)(13,14)(16,21)(19,22,24,23)(25,26,29,32,28,30,36,38)(27,33,35,40,34,31,37,39);(1,3,4,8)(2,5,6,7)(10,14)(11,18)(16,20)(19,24)(26,30)(31,33)(32,38)(39,40);(3,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,36)(38,39);(2,6)(5,7)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(26,30)(27,34)(32,38)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,d^c=di,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122461);