#7454515

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1359 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)(3,6)(4,5)(7,8)(25,26,31,37)(27,32,40,50)(28,41,44,34)(29,36,30,35)(33,46,52,43)(38,53,42,48)(39,54,45,49)(47,55,51,56);(2,5)(3,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,27,31,40)(26,32,37,50)(28,42,44,38)(29,45,30,39)(33,51,52,47)(34,53,41,48)(35,54,36,49)(43,55,46,56);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,28,31,44)(26,33,37,52)(27,38,40,42)(29,43,30,46)(32,47,50,51)(34,35,41,36)(39,55,45,56)(48,54,53,49);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,20)(19,24)(25,28)(26,34)(27,38)(29,46)(30,43)(31,44)(32,48)(33,36)(35,52)(37,41)(39,56)(40,42)(45,55)(47,49)(50,53)(51,54);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,29,31,30)(26,35,37,36)(27,39,40,45)(28,43,44,46)(32,49,50,54)(33,41,52,34)(38,55,42,56)(47,48,51,53)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7361711);