#412673

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

772 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)(3,8)(4,6)(5,7)(9,10,13,18,15,20,14,19)(11,16,23,31,24,32,29,21)(12,25,26,17,28,33,27,34)(22,35,36,30,37,39,38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,31)(19,21)(20,32)(25,35)(26,36)(27,38)(28,37)(33,39)(34,40);(1,4,5,3)(2,6,7,8)(9,10)(11,21)(12,17)(13,19)(14,18)(15,20)(16,29)(22,35)(23,32)(24,31)(25,26)(27,33)(28,34)(30,38)(36,40)(37,39);(9,12,14,27,15,28,13,26)(10,17,18,33,20,34,19,25)(11,22,29,38,24,37,23,36)(16,30,31,39,32,40,21,35)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=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, 319869);