#146406

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#381
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10714
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,12,7,13,26,36,8,11,6,27,25,35,28,51)(3,14,17,38,19,39,45,59,16,9,18,30,43,31,46,55)(4,20,21,33,22,47,48,57,23,10,24,32,49,34,50,56)(15,29,41,52,42,54,61,63,40,37,44,53,60,58,62,64);(1,3,6,18,7,19,28,46,8,16,5,17,25,43,26,45)(2,9,12,30,13,31,36,55,11,14,27,38,35,39,51,59)(4,15,21,41,22,42,48,61,23,40,24,44,49,60,50,62)(10,29,33,53,34,54,57,64,20,37,32,52,47,58,56,63);(1,4)(2,10)(3,15)(5,24)(6,21)(7,22)(8,23)(9,29)(11,20)(12,33)(13,34)(14,37)(16,40)(17,44)(18,41)(19,42)(25,49)(26,50)(27,32)(28,48)(30,53)(31,54)(35,47)(36,57)(38,52)(39,58)(43,60)(45,62)(46,61)(51,56)(55,64)(59,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gi,d^b=di,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53602);