#7202747

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,29)(20,34)(21,28)(22,24)(23,27)(25,33)(26,36)(30,40)(31,45)(32,38)(35,41)(37,42)(39,47)(43,44)(46,48);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,15,16,14)(18,24)(19,30)(26,34)(27,28)(31,44)(32,33)(41,42)(46,47);(3,7)(5,8)(10,11)(14,15)(17,19,22,30)(18,25,24,38)(20,31,36,44)(21,33,23,32)(26,39,34,48)(27,29,28,40)(35,47,37,46)(41,45,42,43);(2,6)(3,7)(9,12,13,16)(10,14,11,15)(17,20,22,36)(18,26,24,34)(19,31,30,44)(21,37,23,35)(25,39,38,48)(27,42,28,41)(29,43,40,45)(32,47,33,46);(3,7)(5,8)(10,11)(14,15)(17,21,22,23)(18,27,24,28)(19,32,30,33)(20,35,36,37)(25,40,38,29)(26,41,34,42)(31,46,44,47)(39,43,48,45)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7109943);