#126831

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#102
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(1,2)(3,4)(5,6,9,12,10,13,8,18)(7,14,17,19,15,20,16,11)(21,22)(23,34)(24,29)(25,31)(26,30)(27,33)(28,36)(32,40)(35,48)(37,42)(38,49)(39,46)(41,47)(43,51)(44,52)(45,50);(2,4)(5,7)(6,11,13,19)(8,16)(9,17)(10,15)(12,14,18,20)(21,23,26,37,24,35,39,50,27,38,41,52,40,51,25,36)(22,28,31,43,32,44,47,49,33,45,46,48,29,42,30,34);(7,15)(11,19)(14,20)(16,17)(21,24,27,40)(22,29,33,32)(23,35,38,51)(25,26,39,41)(28,42,45,44)(30,46,47,31)(34,48,49,43)(36,37,50,52)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34027);