#388992

order	512
nilpotency class	3
isoclinism	isoclinic class of #388990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14533
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fhi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296188);