#237885

order	512
nilpotency class	3
isoclinism	isoclinic class of #232331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=hi,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145081);