#488585

order	512
nilpotency class	4
isoclinism	isoclinic class of #488539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395781);