#127670

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
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	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 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)(7,14)(8,12)(9,18)(10,13)(11,15)(16,19)(17,20)(21,22)(23,32)(24,36)(25,30)(26,41)(27,31)(28,34)(29,39)(33,48)(35,49)(37,44)(38,47)(40,51)(42,46)(43,50)(45,52);(5,7)(6,11,13,20)(8,17)(9,15)(10,16)(12,14,18,19)(21,23,25,35,27,33,26,42)(22,28,30,44,31,43,41,45)(24,37,38,50,40,52,39,34)(29,46,36,32,47,49,51,48);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,20,14)(23,33)(28,43)(32,48)(34,50)(35,42)(37,52)(44,45)(46,49)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=cgh,d^b=dh,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34866);