#152135

order	512
nilpotency class	5
isoclinism	isoclinic class of #152133
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12707
commutator	#117	C₂²×C₄
Frattini, cosocle	#535	C₂³
prime radical	#61637	C₂
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3487 ◁ G
LECS	G ▷ #535 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

658 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,13,17,38)(4,9)(5,21,22,54)(6,24,26,11)(7,12)(8,15,28,44)(10,23)(14,40,43,61)(16,34,25,39)(18,47,48,53)(19,50,52,32)(20,33)(27,41,56,62)(29,45,30,46)(31,49)(35,59)(36,55)(37,51,60,42)(57,63,58,64);(1,3)(2,8)(4,14)(5,15)(6,25)(7,16)(9,27)(10,34)(11,28)(12,30)(13,36)(17,26)(18,41)(19,51)(20,42)(21,39)(22,46)(23,44)(24,29)(31,37)(32,56)(33,58)(35,40)(38,54)(43,52)(45,55)(47,60)(48,64)(49,62)(50,57)(53,61)(59,63);(1,4,7,20)(2,9,12,33)(3,14,16,42)(5,18,22,48)(6,19,26,52)(8,27,30,58)(10,35,36,31)(11,32,24,50)(13,37,34,40)(15,41,46,64)(17,43,25,51)(21,53,54,47)(23,49,55,59)(28,56,29,57)(38,60,39,61)(44,62,45,63)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,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, 59331);