#488700

order	512
nilpotency class	4
isoclinism	isoclinic class of #488699
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×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(18,23)(19,27)(20,29)(21,30)(24,31)(25,28)(26,32);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18,22,25)(19,28,29,23)(20,24,27,32)(21,31,30,26);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(19,29)(20,27)(23,28)(24,32);(2,8)(6,13)(7,14)(12,15)(17,19,21,20,22,29,30,27)(18,24,26,28,25,32,31,23)

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395896);