#7211740

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=fi,e^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=ehi,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, 7118936);