#179735

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86931);