#107025

order	512
nilpotency class	3
isoclinism	isoclinic class of #107018
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

916 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14221);