#7205297

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 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)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,32)(28,34)(29,35)(30,33)(31,36)(37,45)(38,44)(39,47)(40,48)(41,50)(42,46)(43,49)(51,55)(52,54)(53,56);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,37)(29,42)(30,38)(31,39)(33,44)(34,45)(35,46)(36,47)(40,52)(41,53)(43,51)(48,54)(49,55)(50,56);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(29,30)(32,44)(34,48)(35,36)(38,42)(40,43)(45,54)(46,47)(49,50)(51,52)(55,56);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,31,41)(26,34,36,50)(27,37,39,53)(29,43,30,40)(32,45,47,56)(33,48,35,49)(38,52,42,51)(44,54,46,55);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(25,29,31,30)(26,35,36,33)(27,38,39,42)(28,40,41,43)(32,44,47,46)(34,48,50,49)(37,51,53,52)(45,55,56,54)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=fi,e^2=i,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^c=ei,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, 7112493);