#235023

order	512
nilpotency class	3
isoclinism	isoclinic class of #235012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35329	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dhi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142219);