#421902

order	512
nilpotency class	3
isoclinism	isoclinic class of #419041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329098);