#389021

order	512
nilpotency class	3
isoclinism	isoclinic class of #389018
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,f^2=i,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^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285845, #72286090, #72286134
maximal normal subgroup of	#72285845, #72286090, #72286134

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296217);