#6490820

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1529 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,23,28,24,29,22,27)(19,30,33,40,34,43,38,25)(20,35,36,42,21,26,37,41)(31,45,47,39,32,44,46,48);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(17,19)(18,25)(20,31)(21,32)(22,33)(23,38)(24,34)(26,39)(27,30)(28,43)(29,40)(35,48)(36,46)(37,47)(41,45)(42,44);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,20)(18,26)(19,31)(21,24)(22,37)(23,36)(25,39)(27,42)(28,41)(29,35)(30,44)(32,34)(33,47)(38,46)(40,48)(43,45);(9,12,14,13)(10,15,11,16)(17,21,24,20)(18,26,29,35)(19,32,34,31)(22,36,23,37)(25,39,40,48)(27,42,28,41)(30,44,43,45)(33,46,38,47);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(17,22,24,23)(18,27,29,28)(19,33,34,38)(20,36,21,37)(25,30,40,43)(26,41,35,42)(31,46,32,47)(39,45,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398016);