#214804

order	512
nilpotency class	3
isoclinism	isoclinic class of #214704
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#36020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1089 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)(4,7)(6,8)(9,10,13,18,14,17,26,32)(11,19,22,33,21,15,25,28)(12,16,23,31,24,30,37,40)(20,27,35,38,34,29,36,39);(9,11,14,21)(10,15,17,19)(12,20,24,34)(13,25,26,22)(16,29,30,27)(18,33,32,28)(23,36,37,35)(31,38,40,39);(2,5)(3,8)(6,7)(9,12,14,24)(10,16,17,30)(11,20,21,34)(13,23,26,37)(15,27,19,29)(18,31,32,40)(22,35,25,36)(28,38,33,39);(10,17)(13,26)(15,19)(16,30)(22,25)(23,37)(27,29)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122000);