#280892

order	512
nilpotency class	3
isoclinism	isoclinic class of #280883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

478 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188088);