#479777

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

617 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386973);