#111005

order	512
nilpotency class	3
isoclinism	isoclinic class of #110954
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

734 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18201);