#357542

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

4298 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)(9,10,14,20,15,18,13,19)(11,21,25,32,23,16,24,33)(12,26,29,35,30,17,28,36)(22,31,39,40,38,37,27,34);(2,4)(5,6)(7,8)(9,11)(10,16)(12,27)(13,25)(14,24)(15,23)(17,34)(18,21)(19,33)(20,32)(22,28)(26,40)(29,38)(30,39)(31,35)(36,37);(6,8)(9,12,15,30)(10,17,18,26)(11,22,23,38)(13,29,14,28)(16,31,21,37)(19,36,20,35)(24,39,25,27)(32,40,33,34);(11,23)(16,21)(22,38)(24,25)(27,39)(31,37)(32,33)(34,40)

Polycyclic presentation:

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

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, 264738);