#6368804

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368766
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

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306547, #72306674, #72306696, …
maximal normal subgroup of	#72306547, #72306674, #72306696, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276000);