#221218

order	512
nilpotency class	3
isoclinism	isoclinic class of #221204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3506	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

861 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fhi,g^c=gh,e^d=eh

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, 128414);