#127350

order	512
nilpotency class	4
isoclinism	isoclinic class of #127265
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

682 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)(3,7,8,5)(4,6)(9,10)(11,21,24,35)(12,16)(13,29)(14,20)(15,23,32,25)(17,27)(18,36)(19,28)(22,37,26,40)(30,34)(31,38,33,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,25,12,26,27,38,14,24,30,23,28,22,36,39)(10,15,18,21,19,31,34,37,20,32,17,35,16,33,29,40);(9,12,14,28)(10,16,20,19)(11,22,24,26)(13,27,30,36)(15,31,32,33)(17,34,18,29)(21,37,35,40)(23,38,25,39)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=ghi,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ci,d^b=di,e^b=eh

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, 34546);