#468881

order	512
nilpotency class	3
isoclinism	isoclinic class of #454970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 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)(9,10,15,19,13,20,14,28)(11,16,29,33,23,34,24,36)(12,25,30,35,26,18,27,17)(21,32,39,40,22,38,37,31);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,31)(18,32)(19,36)(20,34)(21,26)(25,38)(27,37)(28,33)(30,39)(35,40);(3,7)(6,8)(9,12,13,26)(10,17)(11,21,23,22)(14,27,15,30)(16,31)(18,28)(19,25)(20,35)(24,37,29,39)(32,33)(34,40)(36,38);(1,3,4,7)(2,6,5,8)(9,12)(10,18,20,25)(11,22)(13,26)(14,27)(15,30)(16,32,34,38)(17,28,35,19)(21,23)(24,39)(29,37)(31,33,40,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376077);