#7024188

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

1530 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)(21,36)(22,23)(24,31)(26,37)(28,41)(29,45)(30,38)(32,39)(33,48)(34,44)(35,43)(40,46)(42,47);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,32)(26,36)(30,31)(34,35)(40,42)(45,48);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,31,22,28,37,30)(18,24,26,27,23,38,36,41)(20,33,35,47,32,29,34,46)(25,42,44,48,39,40,43,45);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,20,22,32)(18,25,23,39)(19,29,28,33)(21,35,37,34)(24,40,38,42)(26,44,36,43)(27,45,41,48)(30,47,31,46);(10,14)(13,15)(17,21,22,37)(18,26,23,36)(19,30,28,31)(20,34,32,35)(24,41,38,27)(25,43,39,44)(29,46,33,47)(40,45,42,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=fgi,d^2=f,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=dh,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, 6931384);