#132094

order	512
nilpotency class	4
isoclinism	isoclinic class of #131979
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#540 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39290);