#180316

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

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

(2,6)(3,4)(7,8)(10,15)(11,18)(12,13)(17,20)(19,21)(22,24)(25,26,30,34)(27,35,37,48)(28,32,39,46)(29,41,43,33)(31,42,45,38)(36,49,51,56)(40,53,52,47)(44,54,55,50);(1,2)(3,6)(4,7)(5,8)(9,10,14,17)(11,16,19,23)(12,20,13,15)(18,24,21,22)(25,27,30,37)(26,31,34,45)(28,36,39,51)(29,42,43,38)(32,44,46,55)(33,48,41,35)(40,54,52,50)(47,56,53,49);(3,4)(6,7)(9,11,14,19)(10,16,17,23)(12,18,13,21)(15,22,20,24)(25,28)(26,32)(27,36)(29,40)(30,39)(31,44)(33,47)(34,46)(35,49)(37,51)(38,50)(41,53)(42,54)(43,52)(45,55)(48,56);(1,3,5,4)(2,7,8,6)(11,19)(16,23)(18,21)(22,24)(27,37)(28,39)(31,45)(32,46)(35,48)(38,42)(40,52)(47,53)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=h,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, 87512);