#358413

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1012 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22,25,34)(13,18)(14,17)(16,24,32,26)(19,35,23,39)(20,30)(21,29)(27,33)(28,36,31,37)(38,40);(2,5)(6,8)(9,11,13,20,14,21,27,38)(10,15,18,30,17,29,33,40)(12,23,24,37,25,19,26,36)(16,31,22,39,32,28,34,35);(9,12)(10,16)(11,19)(13,26)(14,25)(15,28)(17,32)(18,34)(20,37)(21,23)(22,33)(24,27)(29,31)(30,39)(35,40)(36,38);(1,3,4,7)(2,6,5,8)(10,17)(15,29)(16,32)(18,33)(22,34)(28,31)(30,40)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265609);