#435541

order	512
nilpotency class	3
isoclinism	isoclinic class of #418863
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

785 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:

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

Polycyclic presentation:

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

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, 342737);