#6484598

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473153
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1619 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6391794);