#304830

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

748 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:

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

Polycyclic presentation:

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

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, 212026);