#6528539

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6435735);