#205925

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

1021 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)(17,18,21,25,22,24,29,19)(20,23,28,31,27,30,32,26);(1,2,4,6,5,8,11,13)(3,7,9,12,10,14,15,16)(17,19,21,24,22,25,29,18)(20,26,28,30,27,31,32,23);(1,3,4,9,5,10,11,15)(2,7,6,12,8,14,13,16)(17,20)(18,23)(19,26)(21,28)(22,27)(24,30)(25,31)(29,32);(3,10)(7,14)(9,15)(12,16)(20,27)(23,30)(26,31)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,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, 113121);