#109643

order	512
nilpotency class	3
isoclinism	isoclinic class of #9345
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^c=eh

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, 16839);