#151772

order	512
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	C₂⁴	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,3,4)(5,6)(7,19)(8,25)(9,31)(10,34)(11,35)(12,18)(13,21)(14,28)(15,41)(16,26)(17,27)(20,45)(22,56)(23,37)(24,57)(29,44)(30,43)(32,61)(33,58)(36,49)(38,50)(39,65)(40,60)(42,59)(46,54)(47,53)(48,52)(51,63)(55,62)(64,67)(66,68);(5,7,9,22,32,51,8,21,27,52,60,68,26,53,41,65,12,20,33,50,10,23,30,49,11,24,28,54,59,67,29,55)(6,13,15,37,42,56,40,45,35,62,25,47,34,46,31,48,18,36,44,63,16,38,14,19,17,39,43,64,61,66,58,57);(7,20)(13,36)(19,45)(21,49)(22,50)(23,51)(24,52)(37,63)(38,56)(39,62)(46,66)(47,64)(48,57)(53,67)(54,68)(55,65)

Polycyclic presentation:

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

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, 58968);