#6529582

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500157
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	#62838	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2731 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:

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

Polycyclic presentation:

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

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, 6436778);