#6368618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368558
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

1514 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)(3,8)(4,9)(5,7)(6,10)(11,16)(12,14)(13,15)(17,18)(19,29)(20,35)(21,26)(22,39)(23,27)(24,33)(25,37)(28,40)(30,42)(31,47)(32,43)(34,44)(36,45)(38,41)(46,48);(2,7)(4,14)(5,16)(8,15)(9,13)(10,11)(17,19,23,32)(18,24,27,44)(20,30,21,38)(22,34,40,33)(25,41,45,42)(26,46,35,47)(28,43,39,29)(31,36,48,37);(1,3)(2,8)(4,11)(5,13)(6,12)(7,15)(9,16)(10,14)(17,20,23,21)(18,25,27,45)(19,30,32,38)(22,36,40,37)(24,41,44,42)(26,39,35,28)(29,47,43,46)(31,34,48,33);(1,4,6,15)(2,9,10,13)(3,11,12,7)(5,8,16,14)(17,21,23,20)(18,26,27,35)(19,31,32,48)(22,36,40,37)(24,42,44,41)(25,39,45,28)(29,47,43,46)(30,34,38,33);(2,10)(3,12)(7,11)(9,13)(18,27)(19,32)(20,21)(28,39)(33,34)(36,37)(41,42)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306633, #72306653, #72306678
maximal normal subgroup of	#72306633, #72306653, #72306678

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275814);