#151994

order	512
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

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×#366
prime radical	C₂³	C₂×Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=gh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ci,d^b=dg,e^b=eg,g^b=gh,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, 59190);