#135771

order	512
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dhi,c^2=eg,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42967);