#152236

order	512
nilpotency class	5
isoclinism	isoclinic class of #149791
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12797
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,15)(4,21,24,37)(5,27)(6,11)(7,13,31,42)(8,14)(9,17)(10,23,38,58)(12,30,40,28)(16,46,48,36)(18,35)(19,44,53,33)(20,45)(22,57,49,32)(25,55,26,56)(29,41,59,39)(34,52,61,50)(43,47,60,63)(51,54,64,62);(1,3,5,17,8,20,6,18)(2,9,11,15,14,35,27,45)(4,22,23,51,26,16,29,47)(7,19,28,50,31,53,30,52)(10,36,37,62,39,32,55,43)(12,33,13,34,40,44,42,61)(21,54,41,57,56,60,38,46)(24,49,58,64,25,48,59,63);(1,4)(2,10)(3,16)(5,23)(6,29)(7,24)(8,26)(9,32)(11,37)(12,21)(13,41)(14,39)(15,43)(17,47)(18,51)(19,48)(20,22)(25,31)(27,55)(28,58)(30,59)(33,60)(34,46)(35,36)(38,42)(40,56)(44,54)(45,62)(49,53)(50,63)(52,64)(57,61)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^c=eh,f^c=fghi,g^c=gi,f^e=fhi,g^f=gi

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, 59432);