#255497

order	512
nilpotency class	3
isoclinism	isoclinic class of #247047
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#36444	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162693);