#214801

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	#392	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1059 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)(9,10,13,18,14,17,28,32)(11,19,22,33,21,15,26,30)(12,23,24,34,25,16,27,31)(20,29,37,39,36,35,38,40);(9,11,14,21)(10,15,17,19)(12,20,25,36)(13,26,28,22)(16,29,23,35)(18,33,32,30)(24,38,27,37)(31,40,34,39);(2,5)(3,8)(6,7)(15,19)(29,35)(30,33)(39,40);(9,12,14,25)(10,16,17,23)(11,20,21,36)(13,27,28,24)(15,29,19,35)(18,34,32,31)(22,38,26,37)(30,40,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^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, 121997);