#347881

order	512
nilpotency class	3
isoclinism	isoclinic class of #347878
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255077);