#7144342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=i,e^2=hi,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, 7051538);