#398401

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1611 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)(9,10,14,16)(11,19,23,32)(12,24,26,17)(13,27,20,18)(15,28,29,22)(21,35,37,33)(25,31,34,39)(30,36,38,40);(5,6)(7,8)(9,11,20,28,14,23,13,22)(10,15,27,32,16,29,18,19)(12,21,34,40,26,37,25,36)(17,30,39,35,24,38,31,33);(1,2,3,4)(6,8)(10,16)(15,29)(17,24)(18,27)(19,32)(30,38)(31,39)(33,35);(9,12,13,25,14,26,20,34)(10,17,18,31,16,24,27,39)(11,21,22,36,23,37,28,40)(15,30,19,33,29,38,32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,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, 305597);