#7498622

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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

1297 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7405818);