#152141

order	512
nilpotency class	5
isoclinism	isoclinic class of #152133
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	#61635	C₂
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3485 ◁ G
LECS	G ▷ #536 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 654 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:

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

Polycyclic presentation:

2^9|a^2=f,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,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

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59337);