#127127

order	512
nilpotency class	4
isoclinism	isoclinic class of #11508
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	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=h,b^a=bd,c^a=ch,d^a=dg,f^a=fi,g^a=gi,c^b=ci,e^b=egi,e^d=ei

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, 34323);