#7355347

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#53151	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(2,4)(3,7)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,29,31)(27,35,36,51)(28,33,41,45)(30,43,37,34)(32,46,44,39)(38,50,40,54)(42,56,52,49)(47,55,48,53);(1,2)(3,8)(4,5)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(26,31)(27,36)(33,45)(34,43)(38,40)(39,46)(49,56)(53,55);(1,3)(2,6)(4,8)(5,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,37,46,29,36,30,39)(26,32,43,51,31,44,34,35)(28,40,52,53,41,38,42,55)(33,48,56,50,45,47,49,54);(3,7)(6,8)(11,20)(15,18)(19,21)(23,24)(25,28,29,41)(26,33,31,45)(27,38,36,40)(30,42,37,52)(32,47,44,48)(34,49,43,56)(35,50,51,54)(39,53,46,55);(1,4,5,2)(3,8,7,6)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(26,31)(27,36)(28,41)(34,43)(39,46)(42,52)(47,48)(50,54)

Polycyclic presentation:

2^9|a^2=fhi,c^2=fg,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,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, 7262543);