#296524

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

758 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=i,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 203720);