#152019

order	512
nilpotency class	5
isoclinism	isoclinic class of #152017
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12687
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂×#585	C₄
LCS	G ▷ C₂²×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1309 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 1309 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:

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

Polycyclic presentation:

2^9|a^2=f,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,e^b=ei,f^b=fgi,g^b=gi,h^b=hi,d^c=di,f^d=fhi,g^d=gi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59215);