#374122

order	512
nilpotency class	3
isoclinism	isoclinic class of #374120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14134
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2391 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281318);