#126847

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ch,d^b=di,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, 34043);