#422498

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1229 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=ei,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,g^c=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, 329694);