#146514

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dhi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53710);