#112258

order	512
nilpotency class	3
isoclinism	isoclinic class of #112245
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 19454);