#452794

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

883 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=efg,c^2=efi,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^b=hi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359990);