#347908

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18300	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1101 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)(13,25)(16,26)(17,27)(18,28)(19,32)(20,35)(22,36)(23,31)(29,39)(30,38)(33,37);(9,10,13,15,14,18,25,28)(11,16,20,26,21,30,35,38)(12,22,23,36,24,17,31,27)(19,33,34,39,32,29,40,37);(2,5)(3,8)(6,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,26)(17,29)(18,30)(22,33)(23,34)(24,32)(25,35)(27,37)(28,38)(31,40)(36,39);(9,12)(10,17)(11,19)(13,23)(14,24)(15,27)(16,29)(18,22)(20,34)(21,32)(25,31)(26,37)(28,36)(30,33)(35,40)(38,39)

Polycyclic presentation:

2^9|b^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dh,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, 255104);