#127611

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

786 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)(9,10,13,19,15,21,14,20)(11,22,24,38,26,32,25,39)(12,18,28,37,29,17,30,36)(16,31,33,27,35,40,34,23);(2,6,8,7)(9,11,14,25,15,26,13,24)(10,16,20,34,21,35,19,33)(12,27,28,40,29,23,30,31)(17,22,36,38,18,32,37,39);(1,3,5,4)(2,7,8,6)(9,12)(10,17)(11,23)(13,28)(14,30)(15,29)(16,32)(18,21)(19,36)(20,37)(22,35)(24,31)(25,40)(26,27)(33,39)(34,38)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34807);