#7202554

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,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=eh,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, 7109750);