#94758

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂²	C₄×C₃₂
Frattini, cosocle	C₂²×C₃₂	C₂²
socle	C₂²	C₂²⋊C₃₂
prime radical	C₂³	C₂×C₃₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ci,d^b=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1954);