#258121

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#39548	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=hi,d^2=ehi,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165317);