#233831

order	512
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂×#3150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1524 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:

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

Polycyclic presentation:

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

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, 141027);