#219504

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#496	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126700);