#7490693

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1169 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,35,31,36,29,44)(27,32,38,50,39,51,45,49)(28,40,42,53,43,34,46,33)(37,54,56,52,41,48,55,47);(2,6)(3,4)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,31,39)(26,32,36,51)(28,41,43,37)(29,45,30,38)(33,52,53,47)(34,54,40,48)(35,50,44,49)(42,55,46,56);(1,2)(3,8)(4,7)(5,6)(25,28,31,43)(26,33,36,53)(27,37,39,41)(29,42,30,46)(32,47,51,52)(34,44,40,35)(38,55,45,56)(48,49,54,50);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,37)(29,46)(30,42)(31,43)(32,48)(33,35)(36,40)(38,56)(39,41)(44,53)(45,55)(47,50)(49,52)(51,54);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,24)(22,23)(25,29,31,30)(26,35,36,44)(27,38,39,45)(28,42,43,46)(32,49,51,50)(33,34,53,40)(37,55,41,56)(47,54,52,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,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, 7397889);