#6374425

order	512
nilpotency class	3
isoclinism	isoclinic class of #60753
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^b=di,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281621);