#172240

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
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	#45801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

910 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79436);