#451897

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

823 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)(6,7)(9,10,14,19,15,20,13,17)(11,21,24,33,25,32,22,16)(12,26,28,36,30,34,29,18)(23,39,27,35,40,31,38,37);(1,2,4,5)(3,7,8,6)(9,11,13,22,15,25,14,24)(10,16,19,21,20,33,17,32)(12,27,28,40,30,38,29,23)(18,35,34,39,36,37,26,31);(10,17)(11,22)(13,14)(18,34)(19,20)(21,32)(23,38)(24,25)(26,36)(27,40)(28,29)(35,37);(1,3,4,8)(2,6,5,7)(9,12)(10,18,20,36)(11,23)(13,29)(14,28)(15,30)(16,31,33,39)(17,34,19,26)(21,37,32,35)(22,38)(24,27)(25,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 359093);