#6608870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(11,16)(12,14)(13,15)(17,18,21,25)(19,27,29,23)(20,31,33,46)(22,36,35,26)(24,39,41,34)(28,43,32,47)(30,38,45,42)(37,44,40,48);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,21,29)(18,23,25,27)(20,32,33,28)(22,30,35,45)(24,40,41,37)(26,38,36,42)(31,43,46,47)(34,48,39,44);(1,3,5,4)(2,7,6,8)(10,15)(11,12)(17,20,22,34,21,33,35,39)(18,24,26,31,25,41,36,46)(19,28,30,44,29,32,45,48)(23,37,38,47,27,40,42,43);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(19,29)(20,33)(23,27)(24,41)(30,45)(31,46)(34,39)(38,42);(11,12)(14,16)(19,29)(23,27)(28,32)(30,45)(37,40)(38,42)(43,47)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6516066);