#497327

order	512
nilpotency class	4
isoclinism	isoclinic class of #497323
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	#61053	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#578 ◁ 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)(3,13)(4,10)(5,22)(6,27,28,9)(7,12)(8,15)(11,26)(14,40)(16,45,46,39)(17,37)(18,47)(19,38)(20,50,51,32)(21,36)(23,31)(24,35)(25,33,55,52)(29,44,56,53)(30,43)(34,42)(41,58,62,61)(48,57,63,64)(49,59,54,60);(1,3,7,17)(2,8,12,31)(4,18,21,43)(5,23,26,15)(6,16,28,46)(9,29,27,56)(10,34,36,40)(11,37,22,13)(14,24,42,19)(20,48,51,63)(25,53,55,44)(30,38,47,35)(32,58,50,61)(33,45,52,39)(41,54,62,49)(57,60,64,59);(2,9)(8,29)(10,32)(11,33)(12,27)(13,39)(22,52)(30,57)(31,56)(34,58)(35,59)(36,50)(37,45)(38,60)(40,61)(47,64);(1,4,7,21)(2,10,12,36)(3,14,17,42)(5,24,26,19)(6,20,28,51)(8,30,31,47)(9,32,27,50)(11,38,22,35)(13,40,37,34)(15,43,23,18)(16,41,46,62)(25,54,55,49)(29,57,56,64)(33,60,52,59)(39,61,45,58)(44,63,53,48)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404523);