#428523

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
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

1170 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi,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, 335719);