#6349972

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2793 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)(3,11)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,43)(32,38)(33,40)(34,42)(35,41)(36,39)(37,44)(45,48)(46,47);(2,7)(4,5)(8,15)(9,10)(11,16)(13,14)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,38)(27,29)(28,39)(33,44)(34,45)(35,46)(40,47)(41,48)(42,43);(1,3)(2,8)(4,14)(5,13)(6,12)(7,15)(9,16)(10,11)(17,20)(18,25)(19,30)(21,33)(22,34)(23,35)(24,37)(26,40)(27,42)(28,41)(29,43)(31,44)(32,46)(36,45)(38,47)(39,48);(2,9)(7,10)(8,16)(11,15)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,35,34)(24,38,39,29)(25,40,41,42)(30,44,46,45)(37,47,48,43);(2,9)(3,12)(7,10)(13,14)(17,21,23,22)(18,27,28,26)(19,31,32,36)(20,34,35,33)(24,29,39,38)(25,40,41,42)(30,45,46,44)(37,47,48,43)

Polycyclic presentation:

2^9|d^2=hi,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,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306700, #72306705, #72306708, …
maximal normal subgroup of	#72306700, #72306705, #72306708, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257168);