#244020

order	512
nilpotency class	3
isoclinism	isoclinic class of #233431
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=ef,c^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151216);