#127803

order	512
nilpotency class	4
isoclinism	isoclinic class of #11796
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×D₄	C₂³
socle	C₂²	C₂×#342
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,5)(6,7)(9,10,13,16,14,17,12,22)(11,18,21,23,19,24,20,15);(1,2)(3,7,8,5)(4,6)(9,11)(10,15,17,23)(12,20)(13,21)(14,19)(16,18,22,24);(11,19)(15,23)(18,24)(20,21)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34999);