#395324

order	512
nilpotency class	3
isoclinism	isoclinic class of #394098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1464 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^c=hi,f^d=fi,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, 302520);