#213770

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=gh,g^d=gh

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, 120966);