#6482537

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 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)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,37)(27,33,42,50)(28,43,29,35)(30,36,31,46)(34,45,51,44)(38,52,39,48)(40,49,41,55)(47,54,56,53);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,32,42)(26,33,37,50)(28,38,29,39)(30,40,31,41)(34,47,51,56)(35,48,43,52)(36,49,46,55)(44,53,45,54);(11,17)(14,21)(18,19)(22,23)(25,28,32,29)(26,34,37,51)(27,38,42,39)(30,44,31,45)(33,47,50,56)(35,46,43,36)(40,53,41,54)(48,55,52,49);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,28)(26,35,37,43)(27,39,42,38)(30,44,31,45)(33,48,50,52)(34,46,51,36)(40,53,41,54)(47,55,56,49);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(25,30,32,31)(26,36,37,46)(27,40,42,41)(28,44,29,45)(33,49,50,55)(34,35,51,43)(38,53,39,54)(47,48,56,52)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389733);