#419788

order	512
nilpotency class	3
isoclinism	isoclinic class of #419788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15273
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1535 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)(3,13,17,36)(4,9)(5,21,24,10)(6,11)(7,12)(8,16,29,25)(14,39,42,38)(15,43)(18,31,47,50)(19,32)(20,33)(22,28)(23,51,53,35)(26,37)(27,48,30,41)(34,44)(40,57)(45,54)(46,58,63,64)(49,59)(52,55)(56,62)(60,61);(1,3)(2,8)(4,14)(5,22)(6,25)(7,17)(9,30)(10,34)(11,36)(12,29)(13,37)(15,23)(16,26)(18,45)(19,48)(20,42)(21,43)(24,52)(27,33)(28,35)(31,57)(32,38)(39,59)(40,46)(41,49)(44,53)(47,62)(50,60)(51,55)(54,64)(56,58)(61,63);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,27)(10,31)(11,32)(12,33)(13,38)(15,40)(16,41)(17,42)(21,50)(22,45)(23,46)(24,47)(25,48)(26,49)(28,54)(29,30)(34,60)(35,58)(36,39)(37,59)(43,57)(44,61)(51,64)(52,62)(53,63)(55,56);(1,5,6,23,7,24,26,53)(2,10,11,35,12,21,37,51)(3,15,16,22,17,44,25,52)(4,18,19,46,20,47,49,63)(8,28,13,34,29,55,36,43)(9,31,32,58,33,50,59,64)(14,40,41,45,42,61,48,62)(27,54,38,60,30,56,39,57)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318890, #72318900, #72318981
maximal normal subgroup of	#72318890, #72318900, #72318981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326984);