#152055

order	512
nilpotency class	5
isoclinism	isoclinic class of #152053
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12693
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#60648	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 915 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)(3,14,17,8)(4,19,21,36)(5,11)(6,12,27,10)(7,13)(9,20,33,23)(15,41,43,59)(16,40,47,29)(18,30,26,31)(22,51)(24,37,56,35)(25,38)(28,42,57,64)(32,52)(34,54)(39,62,58,46)(44,49,45,63)(48,60,55,61)(50,53);(1,3,7,18)(2,8,13,31)(4,15,21,45)(5,16,25,48)(6,26,27,17)(9,28,33,58)(10,30,12,14)(11,29,38,61)(19,49,36,59)(20,42,23,46)(22,43,53,44)(24,55,56,47)(32,57,34,39)(35,60,37,40)(41,50,63,51)(52,64,54,62);(1,4)(2,9)(3,15)(5,23)(6,22)(7,21)(8,28)(10,32)(11,36)(12,34)(13,33)(14,39)(16,46)(17,44)(18,45)(19,38)(20,25)(24,54)(26,43)(27,53)(29,59)(30,57)(31,58)(35,51)(37,50)(40,63)(41,60)(42,48)(47,64)(49,61)(52,56)(55,62)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,f^b=fgh,g^b=gi,e^c=ei,f^c=fi,f^d=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, 59251);