#176709

order	512
nilpotency class	3
isoclinism	isoclinic class of #176333
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	#39381	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

859 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)(3,7)(4,6)(5,8)(10,15)(11,16)(12,13)(17,18,22,26)(19,27,28,31)(20,25,21,24)(23,30,32,29);(2,6)(3,4)(7,8)(9,10,14,11)(12,16,13,15)(17,19,22,28)(18,23,26,32)(20,30,21,29)(24,31,25,27);(9,11,14,10)(12,16,13,15)(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,15,11,16)(19,28)(23,32)(27,31)(29,30)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83905);