#7031025

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

1368 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)(19,27)(20,24)(21,33)(22,25)(23,30)(26,34)(28,39)(29,40)(31,38)(32,45)(35,42)(36,43)(37,44)(41,47)(46,48);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(19,28)(23,35)(27,39)(29,41)(30,42)(36,46)(40,47)(43,48);(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,28,34,42)(18,23,26,27,25,35,33,39)(20,29,32,43,31,41,44,48)(24,36,37,40,38,46,45,47);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,22,31)(18,24,25,38)(19,29,28,41)(21,32,34,44)(23,36,35,46)(26,37,33,45)(27,40,39,47)(30,43,42,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,25)(20,31)(23,35)(26,33)(27,39)(29,41)(32,44)(43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=fi,e^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=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, 6938221);