#420823

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1928 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:

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

Polycyclic presentation:

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

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, 328019);