#401019

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308215);