#452920

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=e,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^c=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, 360116);