#7334928

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1408 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,e^2=f,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, 7242124);