#497407

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×#102
prime radical	#61105	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1037 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225388, #72225396, #72225399
maximal normal subgroup of	#72225388, #72225396, #72225399

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404603);