#357554

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

4390 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)(11,20)(12,17)(13,18)(14,27)(15,19)(16,22)(21,36)(23,30)(24,32)(25,34)(26,33)(28,31)(29,37)(35,38)(39,40);(1,2)(3,4)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,37)(25,39)(26,38)(32,35)(33,40)(34,36);(2,4)(9,12,15,26)(10,17,19,33)(11,21,23,38)(13,25,14,24)(16,29,31,40)(18,34,27,32)(20,35,30,36)(22,39,28,37);(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,26,25)(16,30,31,20)(17,32,33,34)(21,37,38,39)(29,35,40,36)

Polycyclic presentation:

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

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, 264750);