#6466880

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^c=ei

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, 6374076);