#206011

order	512
nilpotency class	3
isoclinism	isoclinic class of #205799
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

1043 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:

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

Polycyclic presentation:

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

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, 113207);