#7024446

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1537 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6931642);