#406730

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1214 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224748, #72224750, #72224752
maximal normal subgroup of	#72224748, #72224750, #72224752

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313926);