#128584

order	512
nilpotency class	4
isoclinism	isoclinic class of #128584
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8174
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,13,16,36)(4,9)(5,22)(6,26)(7,12)(8,15,29,23)(10,24)(11,27)(14,39,42,63)(17,45,44,40)(18,46,48,60)(19,49)(20,35)(21,33)(25,38)(28,41,58,54)(30,55,57,43)(31,47,59,50)(32,52)(34,53)(37,56)(51,61)(62,64);(1,3)(2,8,11,30,12,29,26,57)(4,18,20,48)(5,23)(6,17)(7,16)(9,31,33,58,35,59,34,28)(10,36,22,45,37,13,38,40)(14,21,42,53)(15,25)(19,50,51,47)(24,55)(27,44)(32,60,61,39,62,46,49,63)(41,64,54,52)(43,56);(1,4)(2,9)(3,14,16,42)(5,19)(6,21)(7,20)(8,28,29,58)(10,32)(11,34)(12,35)(13,39,36,63)(15,41,23,54)(17,18,44,48)(22,49)(24,52)(25,51)(26,33)(27,53)(30,59,57,31)(37,62)(38,61)(40,60,45,46)(43,47,55,50)(56,64)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,f^2=h,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,h^a=hi,c^b=cgh,d^b=di,e^b=ei,f^c=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, 35780);