#493538

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1786
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3133 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)(3,14)(4,19,22,10)(5,11)(6,25)(7,29)(8,13)(9,17)(12,27)(15,42,45,39)(16,40)(18,41)(20,48,23,35)(21,51,53,56)(24,38)(26,32)(28,57)(30,37)(31,44,47,63)(33,46)(34,58)(36,54,49,50)(43,61,55,62)(52,60,64,59);(1,3)(2,9)(4,15)(5,16)(6,26)(7,30)(8,18)(10,31)(11,32)(12,37)(13,33)(14,25)(17,27)(19,47)(20,43)(21,52)(22,45)(23,55)(24,46)(28,58)(29,34)(35,59)(36,61)(38,40)(39,56)(41,57)(42,51)(44,54)(48,60)(49,62)(50,63)(53,64);(1,4)(2,10)(3,15)(5,20)(6,21)(7,23)(8,22)(9,31)(11,35)(12,36)(13,19)(14,39)(16,43)(17,44)(18,45)(24,50)(25,56)(26,52)(27,54)(28,53)(29,48)(30,55)(32,59)(33,47)(34,60)(37,61)(38,49)(40,62)(41,42)(46,63)(51,57)(58,64);(1,5,8,7)(2,11,13,29)(3,16,18,30)(4,20,22,23)(6,24,28,27)(9,32,33,34)(10,35,19,48)(12,25,38,57)(14,40,41,37)(15,43,45,55)(17,26,46,58)(21,50,53,54)(31,59,47,60)(36,56,49,51)(39,62,42,61)(44,52,63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307089, #72307095, #72307109, …
maximal normal subgroup of	#72307089, #72307095, #72307109, …
maximal quotient of	C₂×#493538, #72320558, #72343272

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400734);