#327738

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,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, 234934);