#283588

order	512
nilpotency class	3
isoclinism	isoclinic class of #283579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3035	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,f^c=fh,g^c=ghi,e^d=eh

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, 190784);