#420822

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^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, 328018);