#498106

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22842
prime radical	#3018	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405302);