#6587712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6494908);