#215187

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,f^2=i,g^2=h,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, 122383);