#394363

order	512
nilpotency class	3
isoclinism	isoclinic class of #394362
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14716
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2539 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)(3,9)(4,20)(5,25)(6,14)(7,12)(8,13)(10,24)(11,26)(15,40)(16,43)(17,33)(18,34)(19,32)(21,49)(22,35)(23,38)(27,39)(28,36)(29,37)(30,42)(31,44)(41,55)(45,58)(46,56)(47,57)(48,53)(50,64)(51,59)(52,54)(60,62)(61,63);(1,3,8,18)(2,9,13,34)(4,21)(5,16,27,45)(6,17,7,19)(10,30)(11,31,37,57)(12,32,14,33)(15,23)(20,40)(22,50)(24,46)(25,43,39,58)(26,44,29,47)(28,42)(35,55)(36,56)(38,49)(41,51)(48,60)(52,62)(53,61)(54,63)(59,64);(1,4)(2,10)(3,15)(5,22)(6,28)(7,24)(8,23)(9,30)(11,35)(12,38)(13,36)(14,20)(16,41)(17,46)(18,21)(19,42)(25,54)(26,53)(27,51)(29,52)(31,55)(32,49)(33,40)(34,56)(37,59)(39,48)(43,63)(44,62)(45,50)(47,61)(57,64)(58,60);(1,5)(2,11)(3,16)(4,22)(6,26)(7,29)(8,27)(9,31)(10,35)(12,25)(13,37)(14,39)(15,41)(17,44)(18,45)(19,47)(20,48)(21,50)(23,51)(24,52)(28,53)(30,55)(32,43)(33,58)(34,57)(36,59)(38,54)(40,60)(42,61)(46,62)(49,63)(56,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319703, #72320000, #72320088
maximal normal subgroup of	#72319703, #72320000, #72320088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301559);