#165071

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=g,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72267);