#175604

order	512
nilpotency class	3
isoclinism	isoclinic class of #175533
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

745 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)(3,5)(4,6)(7,8)(9,10,14,13)(11,15,19,18)(12,16,21,20)(17,23,24,22)(26,31)(27,34)(28,29)(33,37)(35,36)(38,39);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,26,30,33)(27,32,36,40)(28,31,29,37)(34,38,35,39);(2,5)(3,8)(6,7)(9,12)(10,16)(11,17)(13,20)(14,21)(15,22)(18,23)(19,24)(25,27,30,36)(26,32,33,40)(28,34,29,35)(31,38,37,39);(2,6)(5,7)(9,13,14,10)(11,18,19,15)(12,20,21,16)(17,23,24,22)(25,28,30,29)(26,31,33,37)(27,35,36,34)(32,39,40,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 82800);