#7622664

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3480	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,e^a=ei,e^b=ei,e^d=ei

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529860);