#152109

order	512
nilpotency class	5
isoclinism	isoclinic class of #152107
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12696
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#3499	C₂²
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

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

normal subgroups only

This group has 637 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)(3,14,17,8)(4,19,22,39)(5,11)(6,12,29,10)(7,13)(9,21,35,25)(15,44,46,50)(16,43,49,31)(18,32,28,33)(20,53,47,60)(23,52)(24,56,57,38)(26,40,59,37)(27,41)(30,45,61,64)(34,54)(36,58)(42,63,62,48)(51,55);(1,3,7,18)(2,8,13,33)(4,20,22,15)(5,24,27,16)(6,28,29,17)(9,30,35,62)(10,32,12,14)(11,38,41,31)(19,50,39,60)(21,45,25,48)(23,47,55,46)(26,49,59,57)(34,61,36,42)(37,43,40,56)(44,52,53,51)(54,64,58,63);(1,4)(2,9)(3,15)(5,25)(6,23)(7,22)(8,30)(10,34)(11,39)(12,36)(13,35)(14,42)(16,48)(17,47)(18,20)(19,41)(21,27)(24,45)(26,58)(28,46)(29,55)(31,50)(32,61)(33,62)(37,52)(38,60)(40,51)(43,53)(44,56)(49,64)(54,59)(57,63)

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,c^b=ch,d^b=dhi,e^b=ei,f^b=fgh,g^b=gi,e^c=ei,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, 59305);