#107512

order	512
nilpotency class	3
isoclinism	isoclinic class of #107494
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

722 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)(3,11)(4,5)(9,10)(12,13)(14,15)(17,18,22,19,21,26,23,25)(20,27,30,31,28,32,29,24);(1,2,5,9,6,10,4,7)(3,8,12,15,13,16,11,14)(17,19,23,26,21,25,22,18)(20,27,29,31,28,32,30,24);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,32)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14708);