#6365667

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306967, #72306977, #72307019
maximal normal subgroup of	#72306967, #72306977, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272863);