#398392

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
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

1640 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)(9,10,15,21)(11,16,24,33)(12,25,29,36)(13,20,14,19)(17,28,37,27)(18,30,32,23)(22,34,38,35)(26,31,39,40);(5,6,8,7)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,34)(18,19)(20,32)(21,33)(22,27)(25,31)(28,38)(29,39)(35,37)(36,40);(1,2,3,4)(9,12,14,28,15,29,13,27)(10,17,20,36,21,37,19,25)(11,22,30,39,24,38,23,26)(16,31,18,35,33,40,32,34);(5,7,8,6)(9,11)(10,18)(12,22)(13,23)(14,30)(15,24)(16,19)(17,35)(20,33)(21,32)(25,31)(26,27)(28,39)(29,38)(34,37)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,d^2=e,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,f^b=fi,g^b=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305588);