#285776

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192972);