#6348456

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348447
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	C₂×#3544	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2760 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306691, #72306693, #72306721, …
maximal normal subgroup of	#72306691, #72306693, #72306721, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255652);