#7045620

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1288 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=fg,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=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6952816);