#6344003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343312
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

4020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=i,g^2=hi,b^a=bf,e^a=eh,f^a=fh,f^b=fi,g^b=ghi,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251199);