#497351

order	512
nilpotency class	4
isoclinism	isoclinic class of #497339
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#61083	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#577 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1091 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,f^2=i,g^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=fi,g^c=ghi,e^d=ei,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404547);