#7490582

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

1167 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)(25,26,32,45,31,37,30,36)(27,33,46,55,41,50,40,49)(28,35,42,34,29,44,43,51)(38,48,52,47,39,54,53,56);(2,6)(3,4)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,31,41)(26,33,37,50)(28,38,29,39)(30,40,32,46)(34,47,51,56)(35,48,44,54)(36,49,45,55)(42,52,43,53);(1,2)(3,8)(4,7)(5,6)(25,28,31,29)(26,34,37,51)(27,38,41,39)(30,42,32,43)(33,47,50,56)(35,36,44,45)(40,52,46,53)(48,49,54,55);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,31,28)(26,35,37,44)(27,39,41,38)(30,42,32,43)(33,48,50,54)(34,36,51,45)(40,52,46,53)(47,49,56,55);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,24)(22,23)(25,30,31,32)(26,36,37,45)(27,40,41,46)(28,42,29,43)(33,49,50,55)(34,44,51,35)(38,52,39,53)(47,54,56,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,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, 7397778);