#178347

order	512
nilpotency class	3
isoclinism	isoclinic class of #178241
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	C₂×C₄²	C₂⁴
prime radical	#2011	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

823 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=h,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, 85543);