#397033

order	512
nilpotency class	3
isoclinism	isoclinic class of #395252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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)(4,8)(5,6)(9,10)(11,20)(12,25)(13,27)(14,18)(15,19)(16,22)(17,29)(21,35)(23,31)(24,33)(26,28)(30,37)(32,39)(34,38)(36,40);(2,6)(7,8)(9,11,14,22,15,24,25,23)(10,16,12,20,19,31,18,33)(13,21,28,36,29,38,39,37)(17,30,26,34,27,40,32,35);(1,3,5,4)(2,7,6,8)(9,12,25,10,15,18,14,19)(11,16,23,33,24,31,22,20)(13,26,39,17,29,32,28,27)(21,30,37,35,38,40,36,34);(1,4,5,3)(2,8,6,7)(9,13,14,28,15,29,25,39)(10,17,12,26,19,27,18,32)(11,21,22,36,24,38,23,37)(16,30,20,34,31,40,33,35)

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,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, 304229);