#128773

order	512
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#327
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	C₄×#327
prime radical	#3054	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

443 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,d^b=dh,g^b=gi,f^d=fi

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, 35969);