#107673

order	512
nilpotency class	3
isoclinism	isoclinic class of #107658
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,5,6,4,3,7,8)(9,10)(11,15,20,31)(12,22)(13,25)(14,17)(16,24)(18,28)(19,35,37,39)(21,33,26,30)(23,34)(27,32)(29,38,40,36);(5,7)(6,8)(9,11)(10,15,17,31)(12,19)(13,26)(14,20)(16,29,32,40)(18,33,25,30)(21,28)(22,39,34,35)(23,37)(24,38)(27,36);(1,3)(2,4)(5,6)(7,8)(9,12,14,23)(10,16,17,32)(11,19,20,37)(13,27,28,24)(15,29,31,40)(18,34,25,22)(21,38,26,36)(30,39,33,35)

Polycyclic presentation:

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

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, 14869);