#180440

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

691 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,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, 87636);