#327760

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

883 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)(9,10,13,17)(11,14,18,22)(12,15,20,16)(19,21,24,23)(25,26,31,28)(27,32,36,38)(29,33,30,39)(34,37,40,35);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24)(25,27)(26,32)(28,38)(29,37)(30,35)(31,36)(33,40)(34,39);(1,3)(2,6)(4,7)(5,8)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(25,28,31,26)(27,34,36,40)(29,39,30,33)(32,35,38,37);(3,7)(6,8)(11,18)(14,22)(19,24)(21,23)(25,29,31,30)(26,33,28,39)(27,35,36,37)(32,34,38,40)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=i,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, 234956);