#449613

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,f^d=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, 356809);