#6384255

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂²×C₄×D₄
prime radical	#61141	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291451);