#328604

order	512
nilpotency class	3
isoclinism	isoclinic class of #327117
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

733 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)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24)(25,26,29,33)(27,31,36,40)(28,32,30,37)(34,38,39,35);(2,5)(6,8)(9,10,13,15)(11,14,18,22)(12,16,19,23)(17,21,20,24)(25,27)(26,31)(28,35)(29,36)(30,38)(32,34)(33,40)(37,39);(3,7)(6,8)(9,11,13,18)(10,14,15,22)(12,20,19,17)(16,24,23,21)(25,26)(27,34)(28,32)(29,33)(30,37)(31,38)(35,40)(36,39);(1,3,4,7)(2,6,5,8)(10,15)(14,22)(16,23)(21,24)(25,28,29,30)(26,32,33,37)(27,35,36,38)(31,34,40,39)

Polycyclic presentation:

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

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, 235800);