#7024123

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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)(21,23)(22,35)(24,30)(26,34)(28,42)(29,41)(31,37)(32,46)(33,48)(36,43)(38,45)(39,47)(40,44);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,31)(26,35)(30,43)(33,44)(38,39)(41,46);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30,21,28,34,43)(18,24,26,27,23,36,35,42)(20,32,33,47,31,29,44,45)(25,39,40,46,37,38,48,41);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,20,21,31)(18,25,23,37)(19,29,28,32)(22,33,34,44)(24,38,36,39)(26,40,35,48)(27,41,42,46)(30,45,43,47);(10,14)(13,15)(18,23)(20,31)(24,36)(26,35)(27,42)(29,32)(33,44)(45,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=fgi,d^2=f,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^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, 6931319);