#496256

order	512
nilpotency class	4
isoclinism	isoclinic class of #496254
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21750
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#60828	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3499 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

840 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)(3,14,19,43)(4,20)(5,26,28,12)(6,11)(7,13)(8,17,35,29)(9,24,39,22)(10,27,42,61)(15,48,50,63)(16,40)(18,33)(21,56,53,64)(23,37)(25,55)(30,41)(31,45)(32,60)(34,54)(36,49)(38,59,44,58)(46,52)(47,51)(57,62);(1,3)(2,8)(4,21)(5,16)(6,29)(7,19)(9,36)(10,40)(11,43)(12,46)(13,35)(14,45)(15,23)(17,31)(18,28)(20,48)(22,57)(24,60)(25,53)(26,34)(27,54)(30,50)(32,38)(33,42)(37,64)(39,47)(41,56)(44,62)(49,58)(51,59)(52,61)(55,63);(1,4)(2,9)(3,15)(5,24)(6,30)(7,25)(8,32)(10,41)(11,44)(12,20)(13,39)(14,47)(16,51)(17,53)(18,49)(19,50)(21,29)(22,28)(23,31)(26,55)(27,58)(33,63)(34,56)(35,62)(36,43)(37,42)(38,45)(40,48)(46,64)(52,57)(54,60)(59,61);(1,5,7,28)(2,10)(3,16,19,18)(4,22,25,24)(6,27,31,61)(8,33)(9,37)(11,26)(12,45)(13,42)(14,34)(15,49,50,51)(17,52,29,54)(20,44)(21,57,53,60)(23,58,30,59)(32,48)(35,40)(36,56)(38,55)(39,41)(43,46)(47,64)(62,63)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403452);