#474181

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

725 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi,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, 381377);