#6349913

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
prime radical	#62759	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2805 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18,23,28)(19,29,32,38)(20,25,35,41)(21,26,22,27)(24,36,39,31)(30,43,46,47)(33,40,34,42)(37,45,48,44);(4,13)(8,12)(9,16)(11,14)(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)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(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);(2,7)(6,15)(8,12)(9,16)(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);(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|a^2=h,d^2=h,e^2=h,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	#72306705, #72306706, #72306707, …
maximal normal subgroup of	#72306705, #72306706, #72306707, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257109);