#114618

order	512
nilpotency class	3
isoclinism	isoclinic class of #114609
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

810 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)(3,11,4,7)(5,10)(6,9)(8,13,15,12)(14,16)(17,18,22,25,23,19,21,27)(20,28,31,26,32,29,30,24);(1,3)(2,7)(4,6)(5,12)(8,16)(9,11)(10,15)(13,14)(17,19,23,18)(20,29,32,28)(21,25,22,27)(24,30,26,31);(1,4,6,3)(2,8,9,15)(5,12,14,13)(7,10,11,16)(17,20,23,32)(18,24,19,26)(21,31,22,30)(25,29,27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,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, 21814);