#178432

order	512
nilpotency class	3
isoclinism	isoclinic class of #178241
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	#25502	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

932 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)(10,14)(11,12)(15,16)(17,18)(19,27)(20,24)(21,30)(22,25)(23,28)(26,31)(29,32);(1,2,4,6)(3,5,7,8)(17,19,22,20)(18,23,25,32)(21,29,31,28)(24,30,27,26);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,20,22,19)(18,24,25,27)(21,29,31,28)(23,30,32,26);(2,6)(5,8)(9,11,13,12)(10,14,15,16)(18,25)(19,20)(26,30)(28,29)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgi,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, 85628);