#127667

order	512
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

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×D₈
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

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34863);