#113174

order	512
nilpotency class	3
isoclinism	isoclinic class of #113169
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

897 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)(9,10,14,19,15,18,13,30)(11,20,24,35,22,16,23,32)(12,25,28,36,29,17,27,34)(21,37,39,40,38,31,26,33);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,26)(13,24)(14,23)(15,22)(17,33)(18,20)(19,35)(21,27)(25,40)(28,38)(29,39)(30,32)(31,34)(36,37);(3,8)(5,7)(9,12,15,29)(10,17,18,25)(11,21,22,38)(13,28,14,27)(16,31,20,37)(19,36,30,34)(23,39,24,26)(32,40,35,33)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=h,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=ehi,f^b=fh,g^b=ghi,d^c=di,f^c=fh,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, 20370);