#127790

order	512
nilpotency class	4
isoclinism	isoclinic class of #127645
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

425 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)(3,7)(4,6)(5,8)(9,10,13,17,14,20,30,19)(11,21,24,37,22,34,26,40)(12,27,28,16,29,35,39,18)(15,31,33,23,32,38,36,25);(2,6)(3,4)(7,8)(9,11,12,25,13,26,28,38,14,22,29,23,30,24,39,31)(10,15,18,21,19,33,35,37,20,32,16,34,17,36,27,40);(1,3,5,4)(2,7,8,6)(11,22)(15,32)(21,34)(23,25)(24,26)(31,38)(33,36)(37,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=g,e^2=i,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309629, #72309644, #72309653, …
maximal normal subgroup of	#72309629, #72309644, #72309653, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34986);