#413324

order	512
nilpotency class	3
isoclinism	isoclinic class of #398436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
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

1234 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)(11,16)(12,26)(13,27)(14,28)(15,20)(17,18)(19,29)(21,38)(22,40)(23,33)(24,34)(25,32)(30,39)(31,35)(36,37);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,21)(13,22)(14,25)(15,24)(17,35)(18,31)(19,33)(20,34)(23,29)(26,38)(27,40)(28,32)(30,36)(37,39);(1,4,5,3)(2,6,7,8)(9,12,14,18,15,27,29,37)(10,17,19,26,20,36,28,13)(11,21,23,39,24,40,25,31)(16,30,32,38,34,35,33,22);(9,13,14,17,15,26,29,36)(10,18,19,27,20,37,28,12)(11,22,25,35,24,38,23,30)(16,31,33,40,34,39,32,21)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=h,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

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, 320520);