#6351804

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351787
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×#3545	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284167, #72285099
maximal normal subgroup of	#72284167, #72285099

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259000);