#114890

order	512
nilpotency class	3
isoclinism	isoclinic class of #114885
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

794 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22086);