#7622889

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁵
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×Q₈ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ci,g^b=ghi,f^c=fh,e^d=ei,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302929, #72303043, #72303046, …
maximal normal subgroup of	#72302929, #72303043, #72303046, …
maximal quotient of	#72312236, #72312672, #72312828

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530085);