#165676

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=ghi,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,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, 72872);