#7144109

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42800	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7051305);