#128675

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#102
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	C₈×#102
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

323 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320529, #72320704, #72320780
maximal normal subgroup of	#72320529, #72320704, #72320780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35871);