#7144037

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,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, 7051233);