#176827

order	512
nilpotency class	3
isoclinism	isoclinic class of #176333
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

624 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)(12,16)(13,22)(14,17)(15,19)(20,24)(21,23)(25,26,30,27)(28,35,36,40)(29,34,39,32)(31,38,33,37);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,24)(18,22)(25,27,30,26)(28,33,36,31)(29,32,39,34)(35,38,40,37);(1,2)(3,6)(4,8)(5,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,24,18)(25,28)(26,31)(27,33)(29,38)(30,36)(32,40)(34,35)(37,39);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,18,24,23)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,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, 84023);