#215424

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1028 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)(11,17,19,15)(12,20)(13,14)(16,21)(18,22,24,23)(25,26)(27,33,35,40)(28,32)(29,31)(30,34,38,36)(37,39);(1,3,4,8)(2,5,6,7)(10,14)(15,17)(16,20)(22,23);(3,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,27)(26,30)(28,36)(29,35)(31,38)(32,40)(33,39)(34,37);(2,6)(5,7)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(26,31)(30,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi

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, 122620);