#7303770

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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

1394 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7210966);