#357582

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

4302 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)(5,6)(7,8)(9,10,14,19,15,18,13,28)(11,20,23,36,24,16,22,31)(12,17,27,35,26,33,29,34)(21,37,25,32,38,30,39,40);(2,4)(9,11,15,24)(10,16,18,20)(12,25,26,39)(13,23,14,22)(17,32,33,40)(19,36,28,31)(21,29,38,27)(30,35,37,34);(6,8)(9,12)(10,17)(11,21)(13,29)(14,27)(15,26)(16,30)(18,33)(19,35)(20,37)(22,39)(23,25)(24,38)(28,34)(31,40)(32,36);(12,26)(17,33)(21,38)(25,39)(27,29)(30,37)(32,40)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264778);