#127716

order	512
nilpotency class	4
isoclinism	isoclinic class of #7262
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

520 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)(3,5)(4,7)(6,8)(9,10)(11,22,25,37)(12,17)(13,29)(14,32)(15,21)(16,24,34,30)(18,27)(19,31)(20,28)(23,38,39,35)(26,36,40,33);(2,5)(3,8)(6,7)(9,11,14,26,28,39,31,24,15,25,27,40,12,23,13,30)(10,16,19,35,17,33,32,22,21,34,29,38,20,36,18,37);(9,12,15,28)(10,17,21,20)(11,23,25,39)(13,27,31,14)(16,33,34,36)(18,19,32,29)(22,38,37,35)(24,26,30,40)

Polycyclic presentation:

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

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, 34912);