#112340

order	512
nilpotency class	3
isoclinism	isoclinic class of #112331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,e^c=eh,g^c=ghi

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, 19536);