#398549

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1274 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=ef,d^2=ei,f^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^c=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, 305745);