#6481596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
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	C₂²×C₄	C₂⁵
prime radical	C₂×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2898 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,23,26)(19,29,31,43)(20,33,36,44)(21,28,22,27)(24,37,40,32)(25,38,42,35)(30,39,34,41)(45,46,48,47);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,24)(20,34)(21,37)(22,32)(23,31)(25,41)(26,40)(27,43)(28,29)(30,36)(33,45)(35,46)(38,47)(39,42)(44,48);(3,9)(6,13)(10,11)(14,15)(17,20)(18,25)(19,30)(21,35)(22,38)(23,36)(24,39)(26,42)(27,33)(28,44)(29,45)(31,34)(32,46)(37,47)(40,41)(43,48);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(18,26)(19,31)(25,42)(27,28)(30,34)(32,37)(33,44)(46,47);(2,8)(6,13)(7,16)(14,15)(18,26)(24,40)(25,42)(27,28)(29,43)(33,44)(39,41)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388792);