#411647

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
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×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1864 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^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,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318843);