#152144

order	512
nilpotency class	5
isoclinism	isoclinic class of #152136
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12707
commutator	#117	C₂²×C₄
Frattini, cosocle	#536	C₂³
prime radical	#3485	C₂²
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3485 ◁ G
LECS	G ▷ #536 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

647 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fhi,g^f=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59340);