#180015

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#30239	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

820 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)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24)(25,26)(27,30)(28,31)(29,33)(32,36)(34,38)(35,37)(39,40);(1,2,5,8)(3,6,4,7)(9,11,14,21)(10,15,17,23)(12,20,13,19)(16,24,22,18)(25,27)(26,30)(28,35)(29,34)(31,37)(32,39)(33,38)(36,40);(3,4)(6,7)(9,10)(11,15)(12,22)(13,16)(14,17)(18,20)(19,24)(21,23)(26,31)(28,36)(30,37)(32,33)(35,40)(38,39);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)(27,34)(30,38)(35,40)(37,39)

Polycyclic presentation:

2^9|b^2=g,d^2=g,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, 87211);