#283592

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

659 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,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, 190788);