#423566

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330762);