#180300

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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

617 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)(11,18)(12,15)(13,17)(14,23)(16,21)(19,20)(22,24)(25,26,30,28)(27,33,35,40)(29,37,39,32)(31,38,34,36);(3,4)(6,8)(9,11,15,22)(10,16,12,19)(13,21,14,20)(17,24,23,18)(25,27,30,35)(26,31,28,34)(29,38,39,36)(32,40,37,33);(1,2,5,7)(3,6,4,8)(9,12,15,10)(11,19,22,16)(13,17,14,23)(18,21,24,20)(25,28,30,26)(27,34,35,31)(29,32,39,37)(33,38,40,36);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,17,12,23)(11,20,22,21)(16,18,19,24)(26,28)(27,35)(32,37)(36,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87496);