#127400

order	512
nilpotency class	4
isoclinism	isoclinic class of #127217
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

777 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=i,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cgi,e^b=eh,d^c=di,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34596);