#128075

order	512
nilpotency class	4
isoclinism	isoclinic class of #128057
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,f^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,g^a=gi,d^b=dh,e^b=egi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35271);