#152120

order	512
nilpotency class	5
isoclinism	isoclinic class of #149799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2128
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12697
prime radical	#60797	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3455 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,15,18,44)(4,21,24,54)(5,26)(6,12)(7,13,31,40)(8,14)(9,17,34,50)(10,23,37,28)(11,29,41,30)(16,46,49,60)(19,33,20,45)(22,55,52,32)(25,39)(27,42,51,35)(36,57)(38,58)(43,62,53,47)(48,59,64,63)(56,61);(1,3)(2,9)(4,22)(5,27)(6,17)(7,18)(8,20)(10,32)(11,15)(12,33)(13,42)(14,35)(16,24)(19,31)(21,53)(23,47)(25,49)(26,44)(28,48)(29,51)(30,50)(34,40)(36,59)(37,60)(38,55)(39,43)(41,45)(46,61)(52,57)(54,63)(56,64)(58,62);(1,4,8,24)(2,10)(3,16,20,22)(5,28,6,23)(7,25,31,57)(9,32)(11,39)(12,21)(13,38)(14,37)(15,43)(17,48,27,47)(18,52,19,49)(26,54)(29,58,30,56)(33,53)(34,46)(35,60)(36,41)(40,61)(42,55)(44,63)(45,59)(50,62,51,64)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,f^b=fghi,g^b=gi,d^c=di,f^c=fi,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, 59316);