#6368754

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368694
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

1501 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)(3,4)(5,7)(6,9)(8,11)(10,14)(12,15)(13,16)(17,18)(19,31)(20,36)(21,27)(22,30)(23,29)(24,28)(25,35)(26,38)(32,41)(33,47)(34,46)(37,42)(39,44)(40,43)(45,48);(2,7)(3,11)(4,13)(5,14)(8,15)(9,10)(17,19,24,34)(18,25,28,43)(20,37,21,32)(22,40,23,35)(26,42,44,41)(27,45,36,47)(29,46,30,31)(33,38,48,39);(1,3)(2,8)(4,9)(5,12)(6,11)(7,13)(10,15)(14,16)(17,20,24,21)(18,26,28,44)(19,32,34,37)(22,38,23,39)(25,41,43,42)(27,29,36,30)(31,47,46,45)(33,35,48,40);(1,4,6,8)(2,3,9,11)(5,15,14,13)(7,12,10,16)(17,21,24,20)(18,27,28,36)(19,33,34,48)(22,38,23,39)(25,42,43,41)(26,29,44,30)(31,47,46,45)(32,35,37,40);(2,9)(3,11)(7,10)(12,16)(18,28)(19,34)(20,21)(29,30)(35,40)(38,39)(41,42)(45,47)

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,c^b=ch,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

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306633, #72306650, #72306670
maximal normal subgroup of	#72306633, #72306650, #72306670

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275950);