#496288

order	512
nilpotency class	4
isoclinism	isoclinic class of #496285
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21758
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60462	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

996 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403484);