#452186

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1141 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,d^2=fi,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,h^b=hi,e^c=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, 359382);