#327708

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
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	#35238	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1033 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)(3,6)(4,5)(7,8)(9,10,13,18)(11,19,21,15)(12,17,14,16)(20,24,22,23)(28,36)(30,39)(32,37)(34,38);(2,5)(6,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,24)(18,19)(25,26)(27,33)(28,37)(29,31)(30,34)(32,36)(35,40)(38,39);(1,3)(2,6)(4,7)(5,8)(10,16)(12,14)(15,23)(17,18)(19,24)(20,22)(25,27,29,35)(26,30,31,39)(28,38,36,34)(32,40,37,33);(3,7)(6,8)(9,12,13,14)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,31)(30,39)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234904);