#496132

order	512
nilpotency class	4
isoclinism	isoclinic class of #496130
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21697
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33722	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

647 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)(4,20,23,36)(5,24)(6,11)(7,13)(8,17)(9,22,35,33)(10,26)(12,25)(15,37)(16,44,48,63)(18,39)(19,41)(21,34,54,60)(27,28)(29,56,58,61)(30,50)(31,42)(32,51)(38,47,62,57)(40,49)(43,55,53,52)(45,64,59,46);(1,3,15,42,6,18,5,17,7,19,26,50,25,49,27,51)(2,8,28,39,12,31,10,14,13,32,24,40,11,30,37,41)(4,16,43,61,22,47,21,46,23,48,53,56,33,57,54,64)(9,29,52,44,36,59,34,38,35,58,55,63,20,45,60,62);(1,4,7,23)(2,9)(3,16,19,48)(5,21,27,54)(6,22,25,33)(8,29)(10,34)(11,20)(12,36)(13,35)(14,38)(15,43,26,53)(17,46,51,64)(18,47,49,57)(24,55)(28,52)(30,45)(31,59)(32,58)(37,60)(39,44)(40,63)(41,62)(42,61,50,56);(1,2)(3,8)(4,20,23,36)(5,10)(6,12)(7,13)(9,33,35,22)(11,25)(14,17)(15,28)(16,45,48,59)(18,31)(19,32)(21,52,54,55)(24,26)(27,37)(29,57,58,47)(30,49)(34,43,60,53)(38,61,62,56)(39,42)(40,50)(41,51)(44,64,63,46)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403328);