#205931

order	512
nilpotency class	3
isoclinism	isoclinic class of #205761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

989 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=fi,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113127);