#132092

order	512
nilpotency class	4
isoclinism	isoclinic class of #131970
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

646 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,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, 39288);